England manager Gareth Southgate's employers want him to extend his contract beyond the end of the 2020 European Championship.
Southgate's popularity with the fans rocketed last month when England reached the semifinals of the World Cup, where the team was beaten by Croatia.
He was handed a four-year deal by the Football Association in November 2016, after taking over from Sam Allardyce.
"Gareth has been excellent, we'd like him to stay beyond 2020," FA chief executive Martin Glenn said on Wednesday.
"I think we'd both like that but if we talked about it at any length it would then be a contract conversation and he's on holiday now, so we'll talk when he comes back."
Southgate's surge of popularity prompted many to speculate that a Premier League club may come calling for the former centre-back.
Glenn hoped, however, that Southgate's "journey" with the national side would prove to be more of a lure than the riches of England's top flight.
"Benchmarks are always the question and we can never compete with the Premier League in terms of pay -- everyone knows that," he said.
"Gareth is on a journey. He loved the World Cup and he's built his own belief that we can go further and that motivates him and [assistant manager] Steve Holland. One of the most powerful things he said after our defeat to Croatia was we haven't done the job.
"I want to pay my staff competitively so they are not a flight risk, but I also recognise we are a sports governing body and we're not a Manchester United.
"At some point, Gareth may choose that he wants to try something different and we'll find someone else."
Southgate, 47, was previously manager of Middlesbrough and the England Under-21 side.