Confirmation of the Ecuadorean's death could stoke tensions between the neighbours, with diplomatic relations between the countries suspended.
The Ecuadorean's body was initially identified as that of a Colombian rebel, alias Julian Conrado, and brought back to the Colombian capital of Bogota with Reyes' body.
Relatives of a missing Ecuadorean locksmith say they have seen news photos that indicate the body is that of their son.
The family of Franklin Aizalia plans to travel to Bogota in a bid to confirm the body's identity.
On Saturday, Rafael Correa, Ecuadorean president, threatened to launch a new diplomatic offensive against Colombia if DNA tests confirmed Colombian forces killed an Ecuadorean citizen.
Santos on Sunday urged Ecuadorean authorities not to act rashly.
"To President Correa and Ecuadorean authorities: Be careful with letting yourselves act impetuously on behalf of criminals," the defence minister said.
"Take care and corroborate with your own authorities the identities of people."
Santos said Colombian military investigations suggest "Lucho" was romantically involved with Nubia Calderon, alias Esperanza.
Calderon allegedly served as a public ambassador in Ecuador to Farc.
Santos called the Colombian raid on the Farc camp a "legitimate act of war".
Correa previously said he would urge the Organisation of American States (OAS) to "act forcefully" if tests confirm that Colombia killed an Ecuadorean citizen, saying he did not want a precedent set in the region.
Over the weekend, the office of Alvaro Uribe, the Colombian president, issued a statement calling Reyes' camp "a place of terrorists".
But he promised to observe any decisions of the OAS, which is investigating the cross-border raid.
Tensions over the raid were largely defused at a regional summit in the Dominican Republic days after the attack, but Correa has yet to return his ambassador to Bogota.
Four Mexican university students and a Colombian soldier were among those who died in the raid.
Uribe, a close ally of America, says documents seized at the camp from Reyes' computer show that the Farc gave money to Correa's 2006 presidential campaign.
He also claims Hugo Chavez, Venezuela's president and an ally of Correa, planned to give the rebels $300m.