Benitez, who is nearing the end of his contract, has so far been reluctant to enter talks with Mike Ashley over a new deal. But after a string of poor results, there is now the threat he could be dismissed early should things not improve.
With the future of Benitez uncertain, Rodgers has been suggested as a potential replacement.
He is currently at the helm of a similarly under-performing Celtic side in the Scottish Premier League. The Bhoys currently sit just third in the SPL table having suffered from patchy form earlier in their season.

Another ex-Liverpool manager, Rodgers does have a great track record of exceeding expectations in the Premier League, most notably with Swansea. Despite operating on a small budget, Rodgers was able to implement his attractive style of football onto the Welsh club, which brought them great success during his tenure.
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A semi-successful stint with Liverpool followed in which they famously let the League title slip away to Man City in 2014.
But in spite of his reputation, many Newcastle fans are still unhappy with the links.
Negative reactions to Rodgers
The fans adore Benitez, and his sacking would surely be a huge error.
There hasn’t been a manager as popular since Kevin Keegan or Bobby Robson, and you’re more likely to hear Rafa’s name being sang on a match-day than any player.

Naturally many fans have came out in support of him:
https://twitter.com/slduk88/status/1051246133970915328
Some were also critical of Rodgers:
https://twitter.com/Aaron_Wilki/status/1051225975831318528
https://twitter.com/GaryToonArmy69/status/1051070031927357441
Positive reactions to Rodgers
It appears though that many fans are already preparing themselves for life without Benitez.
The Spaniard had pushed back contract talks till after the January Transfer Window, where he would be looking for assurances and financial backing in the transfer market.
But with Ashley’s past transfer dealings well known to all, many are already looking to the future.

Some have outlined Rodgers as a great replacement :
https://twitter.com/Semakassa/status/1051144924505214987
https://twitter.com/gibb2018/status/1051884907314720769