The Frenchman, who was linked with a move to Barcelona last month, joined Inter Milan from Monaco for €40m (£36.1m, $47.5m) two summers ago but has failed to impress at the San Siro, scoring just two league goals in 50 Serie A appearances.
According to Italian newspaper Gazzetta dello Sport, the 24-year-old has told Inter Milan manager Luciano Spalletti that he wants a new challenge, in a bid to force his way back into the France squad ahead of next summer's World Cup.
Inter are expected to demand between €25m and €30m for the midfielder, who has not added to his five caps for Les Bleus since 2015, and Gazzetta expects "the most interesting offers" to come from Premier League clubs.
Liverpool, meanwhile, were linked with a move for the Frenchman during the January transfer window last season, amid rumours Jurgen Klopp was ready to bolster his midfield options.
While Gazzetta earmarked Englgnad as the likeliest destination for the Frenchman, who scored a bizarre own goal against Chelsea in a pre-season friendly last week, the newspaper also pointed out Valencia have expressed an interest.
However, Kondogbia's exit from Italy might not be entirely straightforward, particularly as Inter will be looking to recoup a large chunk of the fee they paid Monaco in 2015 ago.
The Nerazzurri could also reportedly offload Croatian playmaker Marcelo Brozovic, who joined from Dinamo Zagreb in 2015, as they look to raise funds to launch an ambitious bid for Bayern Munich midfielder Arturo Vidal.