According to the likes of BBC Sport, Slaven Bilic's Hammers are in for former QPR midfielder Barton on a free transfer although some doubt has been cast over the move after a backlash from the club's fans, as reported by the the Mirror.
The 32-year-old is currently without a club after being released by the West Londoners in June following their relegation from the Premier League.
Barton has calmed down in the last couple of years on the pitch and could represent an attractive proposition for West Ham's owners as he would be arriving on a free transfer.
That would prove attractive to a lot of clubs but particularly the Hammers who, having spent around £25million this summer - although the net spend is less than £20 million - are said to be running out of money to spend of players.
Chequered past makes things difficult
Barton's chequered past will always make life difficult for any clubs looking to bring him in, but he is very much one of those players who fans tend to love when he is on your side.
Another point worth noting is whether Barton would even get in the West Ham team.
Bilic already has a solid core in centre midfield at his disposal - namely Cheikou Kouyate, Mark Noble, Pedro Obiang, Diego Poyet, Reece Oxford and, to a lesser extent, Kevin Nolan.
So the player he looks to sign will surely challenge for a first team spot.
Song of the summer
That player was supposed to be Alex Song.
After a successful loan spell from Barcelona last season, Song was expected to sign the former Arsenal and Cameroon star permanently this summer.
But Bilic's somewhat drawn-out pursuit of the soon-to-be 28-year-old has had to be put on hold due to a lengthy injury which could sideline the African until the new year.
Co-owner David Gold recently admitted the club was still hopeful of signing Song, as reported by the Independent.
But it may be now that they will need to revisit that in January hence the Hammers seeking alternative options in order to plug the gap.
A Braziliant alternative?
Lucas' future at Anfield has been placed in doubt after he was left out of the match-day squad for Liverpool's 1-0 Premier League win at Stoke City at the weekend.
The midfielder's absence was something of a surprise and Reds boss Brendan Rodgers is said by ESPN to be open to offers for the player before the close of the transfer window.
How do the trio compare?
Barton is the oldest of the three at 32, but would still have something to offer in a holding midfield role.
He undoubtedly comes with the most baggage after a string of past misdemeanours both on and off the pitch.
Nevertheless he has played 354 games in his career for four different clubs scoring 32 goals along the way making him the most prolific of the three potential signings.
Song is five years younger than Barton and with a much better pedigree having played at the top level for the likes of the Gunners and Barcelona winning trophies and even captaining his country at major tournaments.
The ex-Bastia man has played 407 matches in a fine career to date including 49 caps for Cameroon and has scored 11 goals in the process but will have stopped and created many more.
Lucas, who is 28, is a fans' favourite at Anfield where he was signed from Brazilian club Gremio. He has played 373 games for the two clubs and his national side in his career scoring 14 goals in that time.
Talented on the ball Lucas would represent quite a coup for the Hammers who are clearly keen to add some experience to their midfield ranks having also been heavily linked with Esteban Cambiasso earlier in the window.
But which of the trio would you prefer to see Bilic sign?