It can't do any harm. If Salah wants to stay, it would be in a successful team, and Klopp is the best option to make that more likely to continue.
Don't have any insight into why Salah won't sign a contract extension, it is usually about money though. If that is the main stumbling block I trust the club to make the best decision for the long term future of the club. Salah getting a significantly larger deal has knock ons for others at the club beyond him, and could cost the club way beyond the cost of a Salah deal. There is no oil money behind Liverpool, or allegedly dodgy deals in Spain, they have to live within their means. I don't want to get close to going out of business as it was close to doing with the last owners.
Mbappe seems to be linked with Liverpool occasionally, but I saw contract offer from PSG for about 800k a week, 😲 (doesn't matter whether €'s or £'s, that is mental
) close to 2x what Salah wants allegedly, and Liverpool are having anything to do with that. As good as Mbappe could be for Liverpool, and I think he would suit Liverpool, no thanks. A Salah deal needs to be in context of the big picture for Liverpool. They seem to be doing things right atm.
If he doesn't sign I would keep him till the end of his contract, to get him at his peak before the start of the inevitable decline with age. With his contract into the last year, offers for him, from the few clubs that could afford him, and his wages, would not be offering top £, because they would think Liverpool will deal at a lower price to not lose him for nothing. I'd say take the goals now.