Barca came out over the weekend to rubbish reports suggesting they could be tempted to part with the World Cup winner when the transfer window re-opens.
A switch to the Premier League has been widely touted, with Chelsea strongly linked to a move in Wednesday's newspapers, while Arsenal are also said to be among those monitoring Villa's situation in Catalunya.
The prolific frontman still has over a year left to run on his contract at Camp Nou, but he is not considered be a guaranteed starter and has suffered serious injury in recent times.
Barca are not prepared to discuss a deal in the New Year, though, and Villa's agent says that that will mean the 31-year-old forward staying put.
"There is always speculation about David because he is a great forward, but his future is in the hands of Barcelona," Victor Onate told Sky Sports.
"They have a contract and it is important to always respect such documents."