Lease Before Deposit

[Ashley Renders]

Hi everyone, 

I made a super rookie mistake and allowed a tenant to sign a lease before she sent the deposit. She said she would send the deposit later that day via e-transfer, so I sent her the lease to sign after checking her credit and calling her references. Now, she says she only has part of the deposit and wants to pay the rest at the end of the week. It seems like a bad sign. 

She has signed the lease, but we have not signed it as landlords. Are we allowed to entertain other offers? Or are we legally bound to rent her the house if she comes up with the deposit some time between now and her move-in date (Dec. 1). 

Lesson learned! Any advice would be appreciated.