Despaigne would have to obtain a visa, a process that can take 3-4 weeks, and pass a physical before a deal could be officially announced.
Despaigne, who turns 27 this month, defected from Cuba during a tournament in Europe last summer, established residency in Spain and began working out in Mexico.
Despaigne is a right-handed pitcher who might peg best as a long reliever if he reaches the big leagues.
In February, Padres executive chairman Ron Fowler vowed that the organization would be more active in the international market.
"We understand international is very important. We're taking a look at how we can improve in that area, it's important for us to be able to compete in that area," Fowler said.
Corey Brock is a reporter for MLB.com. Keep track of @FollowThePadres on Twitter. This story was not subject to the approval of Major League Baseball or its clubs.