Bony, who joined Swansea last summer for a club-record £12million, is a top target for Tottenham while Liverpool are also interested in the striker.
So far, however, neither club has made a concrete offer for the 25-year-old, who scored 16 goals in the Barclays Premier League last season.
Should the Ivory Coast striker depart, the Swans would be left short of options up front after the club confirmed Michu's switch to Napoli on a year-long loan deal on Thursday.
Swansea are currently on a pre-season tour of America but Bony did not travel with the squad having been granted time off following his involvement at the World Cup.
Bony instead returned to Swansea earlier this week after scoring two goals for his country, who failed to progress beyond the group stages in Brazil.
Liverpool and Spurs are also battling to sign 21-year-old Davies.
Swansea expect transfer talks for the Wales left-back to conclude next week after the club's tour of America ends on Sunday.
Davies played 45 minutes in City's 1-1 draw with Chivas Guadalajara on Wednesday and Garry Monk's side play their final fixture against Minnesota United on Saturday.
Monk would like any deal with Tottenham to include a return for midfielder Gylfi Sigurdsson, who has struggled to establish himself in the Spurs side since joining in July 2012.
Sigurdsson scored seven goals in 18 league appearances for the Swans during a successful five-month loan spell from Hoffenheim in 2012.
It is not believed, however, Swansea will pursue Sigurdsson as a stand-alone transfer should Davies not move to White Hart Lane.
Davies may prefer a switch to Liverpool, who can offer Champions League football and a reunion with Brendan Rodgers, who was in charge at Swansea from 2010 to 2012.
The left-back would also follow Welsh duo Joe Allen, who left the Liberty Stadium in August 2012, and Colin Pascoe, who is Liverpool assistant manager.
Davies signed a professional contract with Swansea in 2011 but was unable to break into Rodgers' first-team.
He penned a new two-year deal in 2012 - the same summer Rodgers departed for Liverpool - and has since gone from strength to strength, earning the club's Young Player of the Year award twice in a row.
Davies made 34 Premier League appearances for Swansea last season and scored two goals.
One player definitely leaving the Liberty is Michu, who arrived in South Wales in July 2012 from Rayo Vallecano, scored 22 goals in all competitions during his first season but struggled to recapture that impressive form last term as injuries took their toll
It is understood Swansea were keen to agree a permanent transfer for the Spaniard, who managed only two goals in 15 Premier League starts in 2013/14.
Napoli president Aurelio De Laurentiis tweeted a message of welcome to Michu on Thursday afternoon.
The physical centre forward proved an instant hit after moving to the Liberty Stadium and an excellent campaign saw him voted Player of the Year by both his team-mates and the club's supporters in May 2013.
Michu was called up to the Spain squad for their World Cup qualifier against Belarus in October but ankle and knee problems have since hampered his contribution for club and country.
He will have the chance to play in the Champions League with Napoli, who finished third in Serie A last term.
The Italian outfit also knocked Swansea out of the Europa League in their last 32 tie in February.
Source : PA
Source: PA