**Assuming that your Lease is still current... and that the Lease term isn't up in less than 60 days:
Nope! Sit on it. Tell them that 20 days is not acceptable (Be REALLY nice and explain it to them exactly as you have done here.) Work with them, but be advised that they're lowballing you on BOTH your time and their money. Your Lease gives you the RIGHT to stay put for at least that 60 days. (Does the Lease tell you when the clock starts? Do you need WRITTEN notice? That would be the usual way.)
So... you can offer to leave in 20 but ask them for more money ($1000 moving costs plus your last months' FULL rent back as an inconvenience?) Can you afford First, last and Security in the new place? If not, that should be a consideration, too.
Don't know where you live (state?) but even in a month-to-month scenario, MOST jurisdictions offer 30 days as reasonable. They should at least meet that threshold.
The other option is that they offer you some reasonable access to another place. If they have multiple rentals and a vacancy, for example, you can ask for $1000 and to ride out the rest of the lease there.
Do NOT let them threaten you with the Security deposit concerns. Again in MOST jurisdictions, that is YOUR MONEY and they're just holding it for you. YMMV. Again, this is where knowing your state would help, and others can chime in.
All else fails, Get an attorney to write them a cordial letter. You don't have to be nasty, but you shouldn't have to bend over backwards for someone else's windfall, either.
Good luck.