The Aston Villa forward is regularly used by Thomas Tuchel and came off the bench in England's dramatic win over Mexico, but interest from elsewhere in Europe has begun to cool.
Bayern Munich have already strengthened in Rogers' position by completing the signing of Ismael Saibari, making an offer for the Villa man unlikely. Paris Saint-Germain, meanwhile, have shown no sign of formalising their interest, with their attention currently on RB Leipzig winger Yan Diomande instead.
In England, Arsenal remain the club most strongly linked with Rogers and continue to view him as their top attacking target, though they are also weighing up a move for Paris Saint-Germain's Bradley Barcola as an alternative. No formal offer has yet been made for either player. Tottenham, who have spent heavily this summer and held an interest in Rogers previously, have likewise stopped short of an approach.
For Rogers himself, the priority is playing Champions League football, a competition he has already qualified for with Villa following their fourth-placed Premier League finish. Villa's stance is clear: they do not want to sell, and any club hoping to prise him away would have to smash the British transfer record of £116million that Manchester City paid Nottingham Forest for Elliot Anderson, itself a deal driven up by the increasingly ferocious market for elite Premier League midfield and attacking talent this summer.
Co-owner Nassef Sawiris is expected to have the final say on any offer that does arrive, and Villa's position has only hardened since Anderson's British-record fee reset the market for players of Rogers' calibre and age.
With Bayern out of the running and Paris Saint-Germain focused elsewhere on Diomande, Villa's asking price and Rogers' own preference for European football together make a summer exit look increasingly unlikely, even as his profile continues to rise on the biggest stage in the game.
Arsenal's interest in Barcola as a parallel option could also complicate matters further, should the Gunners decide to prioritise the PSG forward instead of pushing for a formal Rogers bid before the transfer window closes.