Stuart Pearce has been in interim charge of the team following Fabio Capello's resignation in February over the Football Association's decision to strip John Terry of the England captaincy while he fights a racism charge.
Tottenham manager Harry Redknapp is widely considered to be the favourite for the role but so far no official announcement has been made, with the tournament kicking off in less than two months' time.
Lineker, speaking on the Richard Bacon programme on BBC Radio Five Live today [Friday], said: "What is going on with the FA? When are they going to decide or even approach some form of England manager or let Stuart Pearce know he's got (the job) for the Euros? Somebody has to prepare.
"The only reason I can think of - which I'm sure is the reason - is that they don't want to upset any club's applecart.
"If they were going to go for Harry Redknapp, which we all presume they will, they don't want to upset the end of Spurs' season - but it seems they have already done that now so they might as well get on with it.
"They may think everything is in place and there are no (England) games until the end of the season, the training camp is all sorted, but the new man needs to start thinking and planning his team.
"He needs to start thinking and planning about how other teams play and do his homework on them and things like that, now they're running out of time.
"If they were going to go with anybody, they should have done it by now."