It follows reports that Manchester City boss Pep Guardiola, who is out of contract at the end of the season, has been asked about his interest in the role.
Tuchel, 51, led Chelsea to Champions League glory in 2021 but was sacked the following year.
He has been among the bookmakers' favourites for the England job since Gareth Southgate stood down after England lost to Spain in the Euro 2024 final in July.
Sky Sports said the German, who left Bayern Munich at the end of last season, was in "pole position" to become Southgate's long-term replacement.
The Football Association declined to comment on the reports.
Lee Carsley was appointed as Southgate's successor on a temporary basis in August, initially for the Nations League campaign over three international windows.
However, the 50-year-old, who stepped up from his role as under-21s manager, has since given mixed messages about whether he wants the job on a permanent basis.
Carsley, who has overseen three wins and a defeat, believes the job should go to the best candidate, regardless of nationality.
England have twice before been managed by foreign coaches, with Sven-Goran Eriksson and Fabio Capello taking charge.
Carsley's own chances of landing the job full time were severely dented by a 2-1 defeat at home to Greece last week after he named an experimental line-up.
"We've seen in the past that we've had different nationalities coach the team. The best candidate should get the job," said the interim boss.
"I think we'd be putting ourselves in a corner if we didn't, and we didn't open our minds a bit."
