The ceasefire has been threatened with collapse after 34 people were killed in fresh violence.
Akashi said he offered to host the talks between the Sri Lankan government and Tamil Tiger guerrillas after the two parties failed to agree on an acceptable venue to discuss the salvaging of their Norwegian-brokered ceasefire.
"Japan would be delighted if the parties agree to Japan to have the talks," Akashi said on Sunday at the end of a five-day visit during which he met President Mahinda Rajapakse and other leaders.
Earlier, the Tigers wanted the talks held either in territory held by them or at an overseas venue, while the government insisted that negotiations should be within the island.
Norway proposed the island's only international airport as a neutral venue but the Tigers rejected it.
Colombo called for a review of the truce after the August assassination of Lakshman Kadirgamar, the foreign minister, which was blamed on the Tigers. A fresh wave of violence in the past week left 34 people dead, including 18 soldiers.
"I have the distinct feeling that we are entering a new phase in the peace process"
Japanese peace envoy
On Sunday, three people, including a policeman, were killed in violence blamed by the military on the Tigers.
Akashi said the security situation was worsening, but he was hopeful that the peace process could be salvaged.
"The situation in terms of security is clearly deteriorating and it is causing considerable concern to all, including the international community," Akashi said. "However, the situation is not hopeless."
A new phase
"It contains elements of hope. I have the distinct feeling that we are entering a new phase in the peace process."
He said the situation had the potential of "suddenly escalating".
Akashi, who helped raise $4.5 billion in foreign aid pledges in support of the peace process in June 2003, said the island's key aid backers would meet in Brussels on 19 December to review the peace process.
He said he did not travel to the rebel-held regions during his latest visit because he wanted to respect the wishes of the government, which was engaged in a "comprehensive review" of its relations with the Tigers.
The Tamil Tigers are fighting for
an independent homeland
However, he said Japan hoped it would be able to make direct contact with the Tigers in a few weeks to keep up a dialogue.
Formal peace talks have been on hold since April 2003, a month after the sixth and last round of direct negotiations held in Hakone, Japan.
Diplomatic efforts to revive them have been inconclusive, but the Tigers agreed to separate talks on the ceasefire.