Turkey shot down the Russian plane on Tuesday, saying the plane had flown into its airspace.
Lieutenant General Sergei Rudsky said their information suggests that one of the two pilots was killed by rebels after being shot while parachuting from the downed plane.
Videos on social media, that could not be independently verified, appeared to show rebel fighters firing at the pilot as he descended. Russian state media broadcast images of the rebels standing over the pilot's body.
The fate of the other pilot is unclear.
Lieutenant General Rudsky accused Turkey of failing to contact the pilots before firing.
He said Russia is severing all defence contact with Turkey in the aftermath of the incident.
Calls for calm over downed Russian jet
Obama and Hollande also urged Russia and Turkey not to let the situation escalate after Turkey, a NATO ally, said it shot down a Russian warplane near the Turkish-Syrian border after it repeatedly violated Turkish air space.
Obama said the United States did not have enough information yet to form conclusions about the incident, but added similar confrontations could be avoided if Russia stopped attacking "moderate" Syrian rebels who are battling forces loyal to the government of President Bashar al-Assad.
"This points to an ongoing problem with the Russian operations in the sense that they are operating very close to a Turkish border and they are going after moderate opposition that are supported by not only Turkey but a wide range of countries," Obama said.
If Russia directs its energies toward Islamic State forces, "some of those conflicts or potentials for mistakes or escalation are less likely to occur," Obama said.
Russia is supporting Assad's government. Western nations insist Assad needs to step down for peace to take hold in Syria.
"We must prevent an escalation. That would be extremely damaging"
US officials said the Russian aircraft's entry into Turkish air space lasted seconds. Russia condemned the Turkish action, vowing severe consequences.
"We must prevent an escalation. That would be extremely damaging," Hollande said.
Obama said: "Turkey, like every country, has a right to defend its territory and its air space," adding he expected to speak to Turkish President Tayyip Erdogan in coming days.
Obama and Hollande spoke at a White House news conference as the French president visited Washington to coordinate a stepped-up military campaign against Islamic state following the Nov. 13 attacks in Paris that killed 130 people.
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Turkey shoots down Russian warplane
As the two men were talking, French and US jets destroyed an Islamic State command center near Mosul in Iraq, a French official said.
Hollande will meet Russian President Vladimir Putin on Thursday in Moscow as part of a series of talks to increase international pressure on Islamic State after the Paris attacks.
"I will ask President Putin, as I've done before ... that the strikes must be against Daesh (Islamic State), against terrorism," Hollande said.
France has stipulated that for closer coordination with Russia to occur, Moscow must stop Assad from bombing civilians, focus its strikes solely on Islamic State and similar groups and commit seriously to finding a political solution without Assad.
A second senior French official said Paris was under no illusion that getting Putin to coordinate more closely would be easy.
"We feel more like working with him than the United States does, but we reached the conclusion that we had to," the official said. "We have to try and get him on side."
Hollande said France was already coordinating with Russian ships in the eastern Mediterranean since the French Charles De Gaulle aircraft carrier arrived in the region on the weekend.
NATO backs Turkey over Russian plane
NATO Secretary-General Jens Stoltenberg offered support for Ankara on Tuesday after Turkey shot down a Russian warplane near the Syrian border and rejected any suggestion that the incident happened outside of Turkey's borders.
Asked if there was any possibility that the plane was actually shot down inside Syria, Stoltenberg said: "The allied assessments that we have are consistent with the information we have from Turkey," he told a news conference.
"We stand in solidarity with Turkey and support the territorial integrity of our NATO ally," Stoltenberg said.
Australia backs calls for calm over jet strike
Both sides of Australian politics are calling for calm after the shooting down of a Russian warplane by Turkey on the Syrian border.
Foreign Minister Julie Bishop says a clear and precise analysis is needed of Turkey's shooting down of a Russian military jet on the Syrian border.
"There will not be any military retaliation, but clearly there must be an investigation as to what occurred," she told reporters.
Ms Bishop said the incident underscored the complexity of the situation in Syria and Iraq.
Both sides of Australian politics are backing international calls for calm over the incident, which has infuriated Russia.
US President Barack Obama said Turkey had "a right to defend its territory and its airspace" but cautioned against any escalation, while NATO chief Jens Stoltenberg also called for calm.
Cabinet minister Josh Frydenberg agreed, telling Sky News: "That's absolutely right."
Mr Frydenberg described it as the "most serious incident between NATO and Russian forces in nearly half a century".
With so many aircraft from different nations over the region there was a chance of miscalculation, he said.
"I'm sure this story is just at its beginning.
"It shows how fluid, how dynamic, how dangerous the situation is."
Opposition foreign affairs spokeswoman Tanya Plibersek says Australia should not be talking about backing one country over the other.
"We need to resolve the situation so that there isn't this sort of confusion in the future," she told reporters in Canberra.
"All countries involved must work to de-escalate tensions and to improve the communications between militaries."
Turkey and Russia needed to keep lines of communication clear and co-ordinate their efforts against Islamic State, Ms Plibersek said.
Crossbench senator Jacqui Lambie believes the incident is a game-changer and is worried the situation will worsen.
"I do not see President Putin sitting back and taking this, I believe it is going to get very nasty in the next 48 hours," she told reporters.
The senator is calling on the government to go back to the drawing board and come up with a new plan for its role in the conflict.
"Unless we go in there heavy handed and open a dialogue with President Putin, then please withdraw our troops out of there, get us out of there now."
