Spalletti repeated his denial that there had been any physical confrontation but added that the 39-year-old's long history at the club did not entitle him to special treatment.
Totti, in his 24th season at his only professional club, has been used sparingly since Spalletti replaced Rudi Garcia in January.
He has played only eight Serie A matches this season and his late equaliser in Sunday's 3-3 draw at Atalanta was his first goal since September.
Totti had already complained about his lack of playing time in a television interview in February and was left out of the squad for the subsequent game, and there were reports of another rift after Sunday's match.
"It was written that Totti and I had a scuffle in the tunnel, but I didn't meet any players in the tunnel," Spalletti told reporters on Tuesday.
"I spoke with Totti yesterday and we cleared things up. Now we're focussing on Torino."
Roma, who face Torino on Wednesday, have lost only one league match under Spalletti and are third in the table, which would earn them a place in the Champions League playoff round.
Totti predicted last month that he would continue playing for another two years, although that now appears unlikely unless he moves to a different club or league.
"I agree, he saved us, but that's different from saying Roma were just Totti," said Spalletti.
"We have a captain and that's Totti, but we also have a team. I'm the coach and I'm here to make AS Roma win. Rules and decisions are part of my job. I play Totti if he can help us to win. And if he scores, I'm happy."
The shaven-headed coach, in his second stint at Roma, denied he was unsentimental. "I've no time for sentiments? Only when it comes to picking the team, because I want to win. But I love my players."
(Writing by Brian Homewood in Berne; Editing by Gareth Jones)
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