Aldridge was reacting to a national radio report in which former West Ham boss Harry Redknapp claimed that his nephew was about to be sold "for somewhere between £10 and 14 million within the next seven days".
When pressed by the interviewer to justify his comment, 'H' replied: "I should know, I'm his uncle!"
Aldridge revealed: "Yes, we've now had a meeting with Frank about his future and he is clearly finding the situation here difficult without his dad and Harry around.
"We told Frank that we would review the situation should an offer come in from another club, but the fact is we haven't received any offers for him. We're still hopeful that he will stay."
Aldridge admits, however, that he "would not be shocked" if a Premiership rival soon made an official offer for the England midfielder, who will be 23 next month.
Leeds United, where Frank's good friend Rio Ferdinand is now playing a starring role, Spurs and Aston Villa have all been strongly linked with Lampard.
But if he does leave, West Ham will not part with one of their top stars and biggest prospects without a fight - and he won't leave on the cheap, vowed Aldridge.
"If we do receive an offer for Frank, we would be looking at the top end of Harry's valuation of the player," Aldridge added. "We think Frank is a future England captain and it would take a very expensive acquisition to replace him."