As @Joe Gore  stated I've always believed as well that as long as you have a secured interest in the property (i.e. a signed purchase contract AND a Earnest Money Deposit) then it is completely assignable as all contracts are unless stated otherwise in the contract (A HUD 5.0 says you cannot which is why I never use one in my transactions).

Deposit or invest in $100,000 increments CDs, money market, etc at that bank.  

Deposits it may put you at risk but you have to judge based on your screening.

Should I ask for a deposit in their daughter's US account?

I would like to move out of the apartment that I am in, and I will need deposit-cash for a home in Odessa.

Require a deposit that's at least the cost of a months worth of rent. 

If you do, I would charge a pet security deposit, a pet fee for processing paperwork (require vet shot records), and monthly pet rent.

Offer a larger deposit. 

However, like @Ellis San Jose mentioned above, you will command more $$$ for your property if it is delivered vacant (unless you are getting top market rent for that unit).If you do sell with tenants, be prepared when the prospective buyer requests rent rolls, estoppel certificates, rent history, existing leases, rent due dates, security deposits, and any other written or verbal agreements.

Assemble a team of individuals who can split following responsibilities: contribute the earnest deposit money, guarantee the loan, find the deal/conduct the diligence, raise the capital and manage the building.