Should I try to negotiate a scholarship?

Last week I was very pleased to get an offer from school A - it's my top choice school - I told them I was very pelased to accept and looking forward to starting etc however I haven't sent back any of the paperwork yet or the deposit - the deadline for all that is late June. I was told that school A would be reviewing scholarships this week / early June

Today I also got an offer from school B which I'm very pleased with although for various reasons school A suits me better. I know that school B will make scholarship decisions by the end of this month

To be honest, I would go to school A whatever as the programme suits my long term goals better, however I'm wondering if it's worth mentioning to school A that I have the offer from school B and am waiting to hear the outcome on scholarships before making a decision? Is this likely to increase the chances of school A offering me a scholarship?

Another option would be to just mention to school A that I have the offer from school B as part of a general email confirming the status of my payment and paperwork etc but not directly mention scholarships. This would be a more subtle approach and would boost my profile as they'd see I'd got into another top school which could help when they come to review scholarship candidates, however may not be as strong as a negotiation tactic as doesn't address the issue directly

I don't want to negotiate too agressively and sour the relationship, however a potential £20k saving is worth a lot to me and figure it's worth a shot if done in the right way?

Any thoughts on how to go about this?