Barcelona reportedly preparing fourth bid of £138 million for Liverpool midfielder Philippe Coutinho

Sky Sports have suggested that Barcelona are readying an improved £138 million bid for Liverpool's Philippe Coutinho despite contrasting reports the night before that they had given up hope of signing the midfielder.

The Catalan giants have seen three bids rejected - the latest of which was worth an eventual £118 million, although only £82 million of that offer was thought to be up-front - since July.

But despite the La Liga outfit's representatives pressuring Coutinho to force through a move, Liverpool have stood firm and stuck to their stance that the Brazilian is not for sale.

Coutinho even submitted an official transfer request on the eve of the club's Premier League opener at Watford earlier this month although Liverpool swiftly rejected it.

Reports from the Liverpool Echo on Tuesday suggested that Barcelona had conceded defeat in their pursuit of Coutinho.

Much was made of Barcelona setting a deadline for their third and final offer for Coutinho - although the Reds had already flat-out rebuffed the offer and reiterated that the playmaker is not for sale.

Liverpool's sporting director Michael Edwards unsurprisingly ignored Barcelona's deadline for the Premier League side to accept their offer by 7pm on Sunday.

But whether that is truly the end of the matter remains to be seen, with Sky now suggesting that Barcelona will "never give up" trying to secure a deal for the 25-year-old.

Suggestions on social media indicated that a fourth bid, worth a total £138 million and made up of £101 million up-front with a further £37 million in add-ons - related to the player's appearances in the Champions League - is imminent.

If true, even that once-again-improved offer is unlikely to alter Liverpool's perspective on the matter. They have long insisted that they will not entertain interest in their prized asset, no matter the fee on offer.

A confusing and convoluted transfer saga

Liverpool's American owners Fenway Sports Group had declared via a public statement hours before news of his transfer request broke that Coutinho is not for sale at any price this summer.

They have stuck by their word - rejecting the third bid which would have made Coutinho the second-most expensive player in history behind Neymar, who moved from Barcelona to Paris Saint-Germain in a deal worth a staggering £198 million.

Should Coutinho stay at Liverpool beyond the closure of the transfer window - as is expected - the club must hope that he follows suit of fellow South American Luis Suárez.

The Uruguayan scored 31 goals in his best-ever season for Liverpool after being denied a move to Arsenal in 2014 and almost fired Liverpool to the league title before leaving for Barcelona the next summer.

Jürgen Klopp has reportedly deemed it too late in the window to find a suitable replacement for Coutinho, while the club also wanted to make a statement that they can keep hold of their star players in the wake of suggestions that they have become a selling club under FSG.

But Liverpool are now set to face the challenge of bringing Coutinho back into the fold despite midfielder badly damaging his relationship with the club and its supporters throughout a complicated saga.

Coutinho only penned a new long-term contract with Liverpool in January, one which contained no release clause and declared himself happy at Anfield.

But after reports that he would only push for a move if a deal could be completed amicably, he has since repeatedly told Liverpool of his intentions to leave with Barcelona convincing him to submit a transfer request and do his all to propel a move.

That has not happened and Liverpool insist it still will not before the closure of the transfer window on Thursday August 31.

Can Coutinho find his way back at Liverpool?

Coutinho is currently sidelined by a back injury and has not trained since the start of August, missing all of the club's first three games of the season.

But he has still attended all of his rehabilitation sessions and the Anfield club fully expect him to remain professional and reintegrate into Klopp's squad if he fails to get his desired move to Camp Nou.

He is not expected to return to the first-team fold fully until after the international break, when he will face Ecuador and Colombia for Brazil. Liverpool travel to title favourites Manchester City in their first game back, when Coutinho could make his eventual return.

The Reds boss will be keen to restore the attacking midfielder to his senior squad as soon as possible given how key Coutinho is to his side. He enjoyed a career-best campaign with 13 goals and seven assists under the German manager last term.

It is in Liverpool's favour that Coutinho will be battling for a spot in Tite's Brazil squad for the World Cup in Russia next summer, meaning the No.10 will have no option but to get back to his best form to be involved in the international tournament.

Barcelona - who have also been chasing Borussia Dortmund winger Ousmane Dembélé - are otherwise reportedly closing in on the capture of Nice midfielder Jean Michaël Seri.