Sources close to Gary Lineker are increasingly confident the row with the BBC will be resolved to his satisfaction in the next 24 hours.
Sky News also understands there could be a statement on the Lineker issue as early as this morning.
The BBC is also expected to announce it is reviewing its social media guidelines following the controversy, and it is believed the sports broadcaster will agree to be more careful about what he tweets, the Daily Telegraph reported.
Gary Lineker-BBC row – follow the latest
It follows a tumultuous weekend for the BBC and its football shows, notably Match of the Day on Saturday evening which had a much shorter airtime of 20 minutes without commentary, pundit analysis, post-match interviews or theme music.
Lineker has been involved in a row over impartiality by comparing the language used to launch a new government asylum policy with 1930s Germany.
He was suspended by the BBC from presenting Match of the Day on Friday after they said in a statement he would “step back” from hosting the weekly football highlights programme.
Sky News learnt that Lineker had refuted this wording and the corporation had instead forced him off air as he was unwilling to apologise for the comments he had made.
Pundits Ian Wright and Alan Shearer both announced they would not appear on Match of the Day, and a plethora of presenters, former footballers and commentators followed suit in not participating in the BBC’s weekend football shows.
Lineker, who has not tweeted or commented publicly since he was suspended by the corporation on Friday, dodged reporters’ questions again as he left his home in southwest London on Sunday to walk his dog.
“I can’t say anything,” he told the media.
Meanwhile, Match of the Day 2 coverage was only 15 minutes long. Jermain Defoe had previously announced he would not appear as a pundit on the show.
On BBC Two, the Women’s Super League Chelsea vs Manchester United match kicked off with no pre-match presentation and no pundits – it aired live using world feed commentary.
BBC Radio 5 Live programme schedules were also affected for a second day, with it replacing much of its usual live sports coverage over the weekend with pre-recorded content.
Chancellor Jeremy Hunt, speaking to Sophy Ridge on Sunday, said he “profoundly” disagrees with Lineker’s comments.
“If you believe in BBC independence, then it’s not for the chancellor or any other government minister to say how these issues are resolved,” he said.
Mr Hunt added: “The central thing that people want to know is that there isn’t any kind of political agenda in the way the BBC goes about its business, which I’m not saying there is, but that is the confidence people need to have.”