TOP STORY: Liverpool weighing move for Norwich's Lewis
The Reds are said to be prepared to part with £10 million to make the deal happen, but they likely will face competition for the 22-year-old Northern Ireland international.
Klopp is said to be impressed by Lewis' "elite" attitude and believes the player will thrive if offered the chance to develop alongside Trent Alexander-Arnold.
Perhaps not coincidentally, Alexander-Arnold and Lewis are both represented by sports agency PLG, as is Robertson, and that connection has put the Premier League champions on the front foot in the race to secure the player's signature.
Lewis can also operate on the left of midfield, so he will bring even more versatility to the Reds' already attacking-minded backline if an agreement can be reached.
Saints to sign Spurs defender Walker-Peters
The two clubs are said to have negotiated a fee in the region of £12m for the transfer of Walker-Peters, who will make his switch to St. Mary's Stadium permanent following a successful loan spell on the South Coast.
Earlier reports suggested that Denmark midfielder Pierre-Emile Hojbjerg is set to head the opposite direction in a separate £15m deal, and while both transfers are still subject to the two clubs agreeing to personal terms, both players are said to be keen to complete their respective moves.
The 23-year-old Walker-Peters made 10 appearances for Saints this season after joining the club in a temporary deal for the second half of the campaign.
It was widely reported that the player had set his heart on staying with Frank Lampard's side, but the Blues are willing to offer him only a two-year contract.
Willian, who turns 32 next week, also had lucrative offers to move abroad, notably from David Beckham's Inter Miami, but he has made no secret of his desire to stay in London and the Premier League.
And according to both the Mirror and the Star, that has put the Gunners, who secured their first trophy under Mikel Arteta by beating Chelsea 2-1 in Sunday's FA Cup final, in the driver's seat to secure the winger's services.
The Athletic writer David Ornstein adds that the club will now make a formal offer at Arteta's request, with Willian ready to snub the chance to earn more elsewhere in favour of a longer contract.
- Leicester City are keen to sign Burnley defender James Tarkowski, but the Clarets are adamant that the England defender will not leave Turf Moor unless his £50m release clause is met. That would mean a big profit for Sean Dyche's side, who paid just £3.5m to lure Tarkowski from Brentford in 2016.
- Elsewhere, the Mail also says winger Ryan Fraser is considering an offer to join Crystal Palace in a deal that would double his Bournemouth salary. Fraser, 26, is a free agent after opting to run down his contract with the Cherries and has been offered £50,000 per week in a bid to tempt him to Selhurst Park.