The 22-year-old was slapped down by Red Bull earlier this month when he suggested he would not be staying for a fourth year at the team owned by the Austrian energy drink manufacturer.
Horner had said that Sainz would be in a Toro Rosso again next year but there has been persistent speculation that the youngster is set to leave.
One German media report last week suggested he could be in a Renault at the next Hungarian Grand Prix as a replacement for Britain's Jolyon Palmer, who has failed to score a point in 10 races.
Horner chose his words carefully in speaking to reporters after Sunday's British Grand Prix but did not categorically rule anything out.
"I don't know where these rumours come from but I cannot believe that these rumours are out there for the Hungarian race," he said.
Horner said there were no offers on the table at present and emphasised that any decision to let Sainz go would have to make financial sense.
"Carlos Sainz has a contract with Red Bull Racing," he said. "There's two years left on that contract.
"If somebody was prepared to make an offer, of course we'd consider it.
"But it would have to have a significant value attached to it because we've invested in Carlos significantly... you're not just going to give an asset away."
Toro Rosso team boss Franz Tost was not immediately available for comment after a race that saw his drivers collide on the opening, with Sainz retiring on the spot and Russian Daniil Kvyat handed a drive-through penalty.
Toro Rosso has historically been a feeder team for Red Bull Racing but there are no vacancies there, with Australian Daniel Ricciardo and Dutch teenager Max Verstappen locked in contractually.
Kvyat's place might appear at risk, with French prospect and 2016 GP2 champion Pierre Gasly eager to make his F1 debut, but Horner sounded supportive of the Russian in the run-up to Silverstone.
"When you look actually at the performances that he's had in detail, he's actually driven pretty well," said Horner last week.
"We have until December 31st to make our minds up for the future but this year for him, when you look beneath the surface, a pretty decent year so far."
(Reporting by Alan Baldwin, editing by Pritha Sarkar)
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