The Three Lions interim boss raised eyebrows when he mentioned that he would 'hopefully' return to his role as Under-21s boss on the back of his side's shock defeat by Greece last week.
His words have left fans confused over whether he wants the role full-time, with just two games remaining of his initially-scheduled interim stint.
Lineker, meanwhile, suggested on his The Rest Is Football Podcast that Carsley could be aware of what his future holds - something he cannot make public.
The Three Lions hotseat is currently held by Lee Carsley, who has raised eyebrows with his comments on the permanent job
'I wonder - I've heard rumours and stuff - whether he knows something about what's happening in terms of they're looking for a new manager,' Lineker said. 'I think they've sounded out coaches.
'That's what the rumours are, that's what we've heard - and I wonder whether he thinks he's not going to get it so therefore he is perhaps protecting himself and wants to get his other job back.
'I don't know, because it seems a little bit defeatist. "They need a winning coach", I mean, look at the two teams who have just won international tournaments. Spain didn't have a guy who had won trophies before, certainly not in league football, Lee Carlsey has won a World Cup [he won the European Championship with England's U-21s]!
'So he has won a trophy, but he's talking about club sides. I wonder whether they have been sounding out people and he's found out and he just doesn't believe he is going to get the job.'
Carsley has won three of his interim games in charge, picking up victories over Ireland and Finland twice, while he also oversaw the Greece defeat during the current international break.
He has also been backed by some of his players to get the permanent job, with Jack Grealish saying of the Greece game: 'Before people were crying out for all attacking players to play and it didn't work. I don't get it, it can happen in games. I love coming here, a top, top manager and I love playing for him.'
Thomas Tuchel, Pep Guardiola and Eddie Howe remain the favourites to get the full-time gig but it is unclear if any discussions have taken place.
It is expected that the new manager will be in their post by March for the fourth international break of the season.