Havana, Cuba (CNN) -- The Colombian president has dismissed as "media spectacle" a decision by Colombian rebels to release an American captive to civil rights leader the Rev. Jesse Jackson.
Jackson said Sunday that FARC rebel leaders have agreed to release U.S. Army veteran Kevin Scott Sutay.
"They have released him already," he said. "But we must go to get him. He's in Colombia in the jungle. That involves the permission of the government of Colombia and the Red Cross and the facilitating our going into the jungle. Because it's a war zone and there needs to be a cease-fire."
FARC guerrillas captured Sutay on June 20 while he was backpacking through Colombia.
Saturday, Colombian President Juan Manuel Santos tweeted, "Only the Red Cross will be authorized to facilitate the delivery of the North American kidnapped by FARC. We will not allow a media spectacle."
A statement on the presidential website went further: "They need now all kinds of media shows to free a detained person after they promised they would never kidnap anyone."
Colombian officials previously said they feared FARC would use Sutay's release to garner media coverage.
'A humanitarian gesture'
Jackson said FARC invited him to Havana, where the group has been holding peace talks with the Colombian government since last October.
"It's a humanitarian gesture of their own volition, demanding nothing in return," Jackson said at a news conference in the Cuban capital Saturday. "I consider it as a gesture to reinforce the peace negotiation."
FARC guerrillas said in a statement they had previously offered to free Sutay but that "the Colombian government did not provide the minimum necessary conditions we need for this procedure."
The presidential website statement said that if the rebels are indeed acting in good faith, "they should release him quickly without any strings attached."
The U.S. State Department said it was aware that Jackson had "expressed a willingness to assist" and that it was in close contact with the Colombian government.
'He is a tourist'
Jackson said he hopes the Red Cross in Colombia will provide a helicopter to remove Sutay from the remote area of the country where he is being held and that the Colombian government will designate the area a cease-fire zone during his release.
Despite the 10-month old peace talks, the Colombian government has resisted calls for a cease-fire with guerrillas, saying the rebels have used previous cease-fires to strengthen their forces.
Sutay served in the U.S. Army from November 2009 to March 2013. He served in Afghanistan for a year starting in November 2010, according to the U.S. Defense Department.
He is from Willow Spring, North Carolina, and won a number of awards, including an Army Commendation Medal and a National Defense Service Medal. The Pentagon lists his rank as private.
Despite Sutay's military background, Jackson said the Army veteran had peaceful motives for visiting Colombia.
"The fact is, he is a tourist," Jackson said.
FARC, whose name in Spanish is the Revolutionary Armed Forces of Colombia, began fighting the government in the 1960s, making it Latin America's oldest guerrilla force.
There have been sporadic peace talks between the rebels and the government since the 1980s. The last attempt failed in 2002.
CNN's Patrick Oppmann reported from Havana and Nelson Quinones from Atlanta. CNN's Jessica King, Laura Koran and Melissa Gray also contributed.