I strongly suspect the link is being driven by his agent rather than us. We were linked with him in January and Klopp made it very clear it was nonsense. Despite that, there would be some logic to the deal.
Yes there are positions that need improving more urgently but the two positions where we're absolutely threadbare are the Lallana/Wijnaldum box to box/attacking midfield roles and on the wing. Other than Mane we have absolutely nobody that offers us any width and pace, and when you've got Coutinho playing on the other wing we really struggle for pace and width in our attacks without him. Similarly, a big reason for our success this season has been our unorthodox midfield 3 - the work rate, energy and creativity of Lallana and Wijnaldum were pivotal to our success early in the season but we have nobody else in the squad that can play those roles. An injury to one of them and we're suddenly having to change how we play.
Oxlade-Chamberlain isn't the ultimate answer to cover/compete with Mane, we need better than him but he could do a job. The midfield role is more interesting though - there aren't many players that have the characteristics to play the two advanced midfield roles Klopp likes but he's somebody that looks like he could play that role. He's got the energy and intensity to get around the pitch and lead our pressing as Lallana and Wijnaldum do, being naturally a forward he has the creativity and he has some experience of playing in midfield too.
Talk of £35m is fanciful to say the least. Even in today's market, with just one year left on his deal and there not likely to be a host of big clubs lining up to sign him Arsenal would do well to get much more than £20m for him. As for his injury record - I've not checked but I've read a few people suggest that this is massively overplayed.