Speculation from the British press early on Tuesday morning had suggested that the latter were once again targeting a swoop for the £50 million-rated Everton starlet who has developed a deserved reputation as one of English football's brightest young talents in recent years.
Now, according to the Daily Star, Liverpool are set to go head-to-head with the reigning Premier League champions for Barkley having first pondered the possibility of such a deal earlier this year.
While Liverpool do appear to maintain a significant interest in Barkley, it remains to be seen if Everton would be willing to offload such a precocious talent to their fiercest rivals.
Quite simply, at this stage, the Toffees and Roberto Martinez are under no pressure whatsoever to sell the 20-year-old.
Barkley's impressively mature decision to put pen-to-paper on a new four-year deal at Goodison Park following a hugely encouraging 2013/14 season - during which the Merseyside native scored six goals in 34 top-flight appearances - amid serious interest from elsewhere means that there is no immediate danger posed at all to Everton with regards to losing his services for a dramatically reduced fee.
Everton's reluctance to part with a player who has already earned nine senior England caps and has subsequently been tipped for an enormous future at both domestic and international level is perfectly understandable, but in the event that they did decide to cash-in, you would surely have to give the edge to City.
Not only do their vast financial reserves mean that they are more likely to be able to pay the sort of inflated price that will inevitably be demanded during any future negotiations, but their status as perennial title contenders - rather than top-four hopefuls - means they offer far less risk as a potential buyer in terms of the possible damage inflicted on Everton's chances of progression.
The new campaign is yet to properly get underway for Barkley, of course, after he sustained a serious knee injury during pre-season training that prompted initial fears that he could miss the rest of 2014.
Such concerns were allayed earlier this month, however, with Martinez himself confirming that Barkley's rehabilitation was progressing ahead of schedule.