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The Germany international is yet to sign a new deal with his boyhood side and Christian Falk has reported both Premier League teams 'want' him.
Bayern will need to make a decision this summer or risk losing their longest-serving star for free in 2023.
Muller is still one of Bayern's key players and has registered an astonishing 18 assists in 20 Bundesliga games this season.
He's also notched six goals with Julian Nagelsmann's men currently six points clear of Borussia Dortmund.
A move for Muller would likely come in the summer given his importance to Bayern's title charge.
Both Everton and Newcastle are locked in a relegation battle after a poor season.
Everton are without a manager after sacking Rafael Benitez and they've won just one game in 14.
The Toffees are lacking creativity in midfield and have been relying on Demarai Gray and Anthony Gordon to play that position.
Newcastle, meanwhile, are in the relegation zone despite their Saudi-led takeover back in October.
They've already signed Kieran Trippier and Chris Wood from Atletico Madrid and Burnley in this transfer window.
Manager Eddie Howe is keen to bolster his squad and the club have made a move for Manchester United's Jesse Lingard in the last 24 hours.
United are said to be holding out for a £12 million survival fee, which the Magpies are reluctant to pay.
Lyon's Bruno Guimaraes is on their radar and reports on Tuesday claimed the two clubs had agreed a fee.
The Ligue 1 outfit have since released a statement denying an agreement has been reached.
It read: "Olympique Lyonnais categorically denies the false information disseminated by many media reporting an agreement between Newcastle and OL for the transfer of Brazilian international midfielder Bruno Guimaraes.
"In this transfer window period where the slightest rumour is often presented as reality, Olympique Lyonnais reminds that only information officially released by the club is to be taken into consideration."
Guimaraes is regarded as one of the best midfielders in France and has been on the radar of Arsenal and Manchester United in the past.
Do you think Thomas Muller will leave Bayern Munich for Everton or Newcastle?
Let us know in the comments.
Featured Image Credit: Alamy
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