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This morning, we were greeted by an exclusive in The Sun (I know, bear with me) saying that Liverpool manager Brendan Rodgers was desperate to buy Arsenal's Theo Walcott. Walcott, like Robin van Persie, has a year to run on his contract, and like Robin van Persie, Walcott presumably put off contract negotiations with Arsenal, hence the situation we're in now. Because of that, Walcott's departure wouldn't be a big surprise if a club like Manchester City or Chelsea put in a bid for Walcott and offered the £100k/week salary he reportedly wants.
Walcott moving to Liverpool, though, would be a surprise. Brendan Rodgers may very well want Theo Walcott, seeing him as similar to Nathan Dyer or Scott Sinclair, but despite Walcott being a boyhood Liverpool fan, moving from Arsenal to Liverpool would be odd.
For one thing, Arsenal are in a much better position to compete for trophies than Liverpool. Liverpool may have won the Capital One Cup last season, but they finished 8th, a full 18 points behind Arsenal, finishing closer to relegated Bolton Wanderers than they were to Arsenal. Walcott would be moving from a Champions League club where he's a usual starter to a non-Champions League club who are very far from competing for major trophies. For Walcott's career, at least, it'd be a backward step.
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Liverpool may offer Walcott more money than Arsenal, but that shouldn't really be a problem. Arsenal make more money in terms of revenue than Liverpool, because of the bigger stadium and their Champions League status, so should easily be able to offer Walcott more money. He'd be worth it too; he was Arsenal's second top scorer and assister, and although he was inconsistent at times, he still remains an important player for Arsenal.
The final reason why this likely won't happen is because the story is from The Sun, who have a reputation that's as solid as a wet rag.
This isn't to say that Walcott won't leave the club; but given where Liverpool are in comparison to Arsenal, that particular rumour seems unlikely to happen.