The 23-year-old has already turned down an £80,000-a-week offer, as he seeks reassurances from his manager that he'll start to be considered as a central striker.
Gunners boss Arsene Wenger is determined to resolve the current stand-off before Christmas, when Walcott will enter the final six months of his existing deal at the Emirates Stadium.
Reds boss Brendan Rodgers is ready to offer the forward an exit-route out of north London, but Anfield may not be Walcott's only option.
A number of other Premier League clubs - including Chelsea and Manchester City - as well as big-spending Ligue 1 giants Paris Saint-Germain, have also been credited with an interest in his signature, and there's a chance he could wait until the summer, when he becomes a free agent, before committing his future.
The fact that Arsenal have been heavily linked with a January move for Crystal Palace winger Wilfried Zaha - also a reported target for Liverpool - hints that the club are already making plans for Walcott's potential departure.
But, that will not rush him into making a decision, as he pushes for the best possible deal to remain in north London.