Adama Barrow told Senegalese radio station RFM: "According to information we received, there is no money in the coffers. It's what we have been told, but the day we actually take office we will clarify all of it."
West African troops entered Gambia's capital Banjul yesterday to cheers from the city's residents.
They were poised to remove Jammeh - who led Gambia for 22 years but had refused to accept defeat in a December election - but he had flown out of Banjul overnight en route to Equatorial Guinea.
Barrow denied that Jammeh had been offered immunity from prosecution in exchange for leaving the country.
"He wanted to stay in Gambia. We said we couldn't guarantee his security and said that he should leave," Barrow said.
Earlier in the day, the African Union and United Nations published a document on behalf of these two organisations and regional bloc ECOWAS.
In it, they pledged, among other things, to protect Jammeh's rights "as a citizen, a party leader and a former Head of State", to prevent the seizure of property belonging to him and his allies, and to ensure he can eventually return to Gambia.
Barrow said the document did not constitute a binding agreement and said that upon initial inspection it appeared Jammeh had looted state resources.
Jammeh's loss in the December poll and his initial acceptance of the result were celebrated across the tiny nation by Gambians grown weary of his increasingly authoritarian rule.
But he reversed his position a week later, creating a standoff with regional neighbours who demanded he step down.
At a press conference in Banjul, Barrow's spokesman Halifa Sallah said a military aide would be sent today to determine if security conditions permit his return.