There was no immediate confirmation of the report, but shortly after it was broadcast, Olli Rehn, the European Union enlargement commissioner, hailed what he called "good progress" towards the formation of a pro-European reform government.
"Serbia's path to the EU will be revitalised immediately" once such a government was formed, he said.
The talks appeared stalled earlier in the week when Kostunica, a moderate nationalist, backed Tomislav Nikolic, the Radical party deputy leader, for the key post of parliament speaker.
A constitutional deadline expires on May 14 and if there is no agreement by then a new election must be called.
The Democrats said Kostunica was trying to blackmail them into joining a coalition on his terms, and accused him of capitulating to the hardliners, seen by the West as heirs of the late nationalist strongman, Slobodan Milosevic.
The European Union condemned the election of Nikolic as a return to "darker days" and urged Kostunica and Tadic to rise above their differences for the good of the country.
Political sources said that as part of the deal between the two leaders, they would unite in parliament to replace Nikolic as parliament speaker.
They had also reached agreement on joint control of the security services.
Tadic had opposed giving full control of the police and secret service to Kostunica's party because he has not arrested Ratko Mladic, a prominent war crimes fugitive.
The EU froze talks on closer ties with Serbia last year due to Belgrade's failure to hand over Mladic, and says he is hiding in the country under the protection of hardliners in the security services.
If a new election were to be called, the campaign would coincide with the loss of Kosovo, whose Albanian majority expects to win independence soon with Western backing.