Yes you can although many realtors will object. You and a seller can agree to anything that is not illegal. The contract, Agreement of Sale, requires consideration which can be paid at any time agreeable to both parties. The most common method is earnest money with the offer a deposit with acceptance and the balance at closing but both parties can agree to something different.
Fran and Mark Redding
Prudential Fox & Roach, Realtors
1010 stony Hill Road
Yardley, PA 19067
Century 21 Sherlock Homes
My web site has lot's of information also : http://www.century21paonline.com
Then there was an investor who said he offered $10 for earnest money and did ok.
Make it a small amount of earnest money and you will not care if you lose it.