I work for Tech Firm A, and was recently recruited by Tech Firm B. Firm B has made me a fairly nice offer. I was planning to accept it, even though it means moving.

My plan had been to accept and make sure everything was settled before informing my manager at Firm A. However, the recruiter made a definite point of telling me to inform her of any other offers I received, which makes me think that perhaps I should tell my current manager about Firm B, in the hopes that Firm A makes a counteroffer. Which I could then present to the recruiter for Firm B, in hopes that they'd up their original offer.

I suppose my current manager could just fire me on the spot (unlikely; he's a good guy), in which case I would have to drastically advance my plans. But are there any other possible downsides to this scenario? Also, is there any sort of reasonable chance that this could lead to a higher salary offer from Firm B?

Both firms are extremely large and well-known. Firm B has given me until Friday to decide, so I don't have a whole lot of time for figuring this out.