The highly-rated midfielder is expected to leave the East London outfit this summer following the departure of his father Frank Lampard Snr and manager Harry Redknapp at the start of last month.
There are a number of clubs interested in signing the want-away 22-year-old ace and Villa are leading the chase to snap him up, but agreeing a fee has proved to be something of a stumbling block in any deal.
Villa have already had a £7.5 million deal rejected by The Hammers who are thought to be keen to get closer to £12 million for the England international.
"There is on-going discussion regarding Frank," said Villa's finance director Mark Ansell. "Our interest is known to West Ham.
The player's agent Steve Kutner has admitted that the ball is in West Ham's court and it is up to them to sell him or persuade him to stay.
"It is now up to West Ham," said Kutner. "Frank is not in the right frame of mind to begin the season at West Ham and that is good news for anyone who wants to buy him."
Meanwhile, Villa's Turkish defender Alpay Olzan has angrily denied reports linking him with a possible move to Arsenal.
Alpay had reportedly claimed that Arsenal had enquired about him, but the player admitted that he was happy to stay at Villa Park.
"I'm really angry to hear these reports, because I did not say that I wanted to leave," Alpay said on the club's official website.
"Everyone at Villa, the manager, the players and most of all the fans, have been very good to me and I am looking forward to playing for Villa again next season.
"I do not want people to think that I am trying to leave. It is not fair to the fans and it is not fair to me. I want to stay and I will stay at Villa Park.
One player, who is looking to leave Villa Park, Gareth Southgate has dismissed reports that he was set to make a move to Middlesbrough.
The England international who has been on the transfer list at Villa for the past year, is believed to be a top target for Boro, but the player insists that he will wait over the summer before deciding his future.
"We are still waiting for a club to make a bid and then take it from there," said Southgate. "But I won't be signing for Middlesbrough today or tomorrow contrary to some reports.
"I'm not worried about the situation. I've waited that long already. It's been a year since I asked for a transfer. I'm just going to enjoy the summer.
"The fact nothing has been sorted out yet won't spoil my summer," added Southgate.
