According to the Daily Mail, the Sun and the Daily Mirror, the Reds have a firm interest in Williams and are in good position to sign the centre-half despite Arsenal also being keen as the Welshman was a Reds fan as a boy.
However, at 28-years-old he does not fit in with the 'invest in youth' policy from Liverpool's American owners.
As the Mail explains: "Williams was a boyhood Liverpool fan and would favour a move to Merseyside but he is likely to cost in the region of £12million and Fenway Sports Group have implemented a strategy that means they do not sign players over the age of 23.
"Given that Williams is 28, Rodgers would have to persuade FSG to be flexible and that element means the transfer might become complex. Liverpool are aiming to bring in two central defenders and Ajax’s Toby Alderweirled is another prime target."
The Mirror also says that Liverpool are keen to sign two centre-backs and Alderweirled is mentioned in their report too.
At 24 Alderweirled is closer to the American owners target age group. Also linked with Liverpool by the paper are 24-year-old Brazilian and Vasco de Gama star Dede and Feyenoord's Stefan de Vrij who is already club captain and a Dutch international despite being just 21.
Again though these deals will be dependent on what exactly the owners offer Rodgers in the summer.
"(Rodgers) won't know for sure which of those candidates he will bid for, until he finalises his summer transfer budget with the club's American owners, and then decides which players will leave Anfield in the summer to raise funds," reports the Mirror.
"Ideally, Rodgers would like at least four new signings, but that would most likely depend on selling players like Andy Carroll, Pepe Reina and Martin Skrtel, to give him a war chest of around £40m with which to do business."
They might not get the money they want for Carroll though with West Ham set to play hardball on the fee for a permanent transfer.
Liverpool value the striker at £17m but the Hammers are only prepared to pay £10m according to the Daily Mail.
His loan move to Upton Park gave West Ham the option to sign him for that £17m fee that Liverpool were looking for but according to the Mail "his injuries and goals return have prompted West Ham to re-evaluate the deal and they will reduce their offer."
The paper reckons "Liverpool are likely to rebuff that as they try to negotiate a deal with Carroll and Newcastle for Hatem Ben Arfa. Yet Newcastle’s valuation of the France midfielder will also make that deal difficult."
Southampton’s Rickie Lambert and Vitesse Arnhem’s Wilfried Bony are listed as alternative targets for West Ham.
To other rumours and Tottenham are ready to approach Everton over a summer deal for contract rebel Leon Osman. The 31-year-old, who has two years left on his current contract, has broken off talks with the Toffees over an extension and Spurs are reportedly ready to pounce. (Metro)
Manchester City have joined Arsenal and Manchester United by expressing interest in Borussia Dortmund midfielder Mario Gotze. The 20-year-old playmaker revealed he is keen to play for United at some stage but Champions League semi-finalists Dortmund value him at around £35m and expect him to stay another season. (Daily Mail)
Former Arsenal midfielder Mathieu Flamini wants to return to the Premier League this summer. Flamini, 29, has had tentative discussions with Aston Villa and Southampton, as his contract with AC Milan finishes at the end of this campaign. (Daily Mirror)
Manchester United are reviving their interest in Anderlecht’s David Henen, 16. Anderlecht want around £3.5million for the forward, who is under contracted until 2015. Henen turned down United last year. (Daily Mail)
Sunderland are looking at Basle defender Aleksandar Dragovic, 22. Scouts watched him score against Spurs on Thursday night, and he interests West Ham and Arsenal as well. Sunderland also want Toulouse winger Franck Tabanou, 24, and Ghana striker Mahatma Otoo, 21. (Daily Mail)
Stoke boss Tony Pulis is sizing up a bid for highly-rated Denmark and FC Copenhagen striker Andreas Cornelius from under the noses of Everton as he seeks to replace the recently retired Michael Owen. The 6ft 4in player is regarded as one of Denmark's rising stars and it would take at least £5m to capture his signature. (Daily Mail)
Finally, QPR are giving a trial to Feyenoord midfielder Brandon Zigo Tichem, 18, a product of the esteemed Alphense Boys club who can also play as a striker. (Daily Mail)