The Chelsea youngster had a recent spell at Crystal Palace on loan cut short.
Bamford played just nine times for The Eagles, mainly as a substitute, before the rest of his season long loan was cut short earlier this week.
He will now be hoping to find a new club in the coming days - with his preference to stay in the Premier League.
Villa are looking for any new firepower. They look certain to go down this season after accruing just the eight points prior to 2016. Bamford would give them another option up front.
Bournemouth are performing better now but the squad is injury prone and Eddie Howe would like Bamford in as insurance.
Meanwhile Swansea are slim in the striker positions - with Eder looking like a flop and Bafetimbi Gomis inconsistent.
Bamford may not have succeeded at Palace but the Chelsea youngster scored 19 goals for Middlesbrough last term and 25 with MK Dons and Derby County in the previous campaign.