The envoy's visit comes as the country's main financial donors voiced concern over mounting bloodshed.
Japan, the United States, the European Union and Norway, have expressed deep concern over Colombo's recent move to scrap the February 2002 truce with the Liberation Tamil Tigers of Eelam (LTTE), saying they feared it would lead to further bloodshed and stall peace efforts.
Sri Lanka announced it would formally pull out of the six-year truce with the Tigers on January 16, saying the LTTE had used the ceasefire as a cover to rearm, recruit and attack troops.
The quartet of financial donors asked the warring sides to protect civilians, allow humanitarian agencies access to affected people and urged Colombo to hammer out a political solution to stem the violence.
They also asked Colombo for access to meet the LTTE members in territories that they control in the north and urged "continued monitoring of the human rights situation by such means as to assure an appropriate role for the UN".
Sri Lanka has repeatedly rejected previous calls for a UN rights mission here and there was no immediate comment from the government.
Analysts have said Akashi's three-day mission might prove fruitless.
Paikiasothy Saravanamuttu, head of the Centre for Policy Alternatives, a Colombo think-tank, said "the government is hell-bent on pursuing its military plans".
"Japan has serious economic clout in Sri Lanka and it will be interesting to see what his message to the government is," Saravanamuttu said.
Akashi's trip follows the killing of D M Dassanayake, the minister for nation-building, on Tuesday, in a bomb attack blamed on the LTTE.
Rajapakse said the killing of Dassanayake, who was given a state funeral late on Saturday, marked a need "to redouble our efforts to rid our country of terrorism".
Fighting in the island's north along the LTTE-held areas of Jaffna, Vavuniya and Mannar has been rising with government forces claiming they now have the upper hand in the decades-old conflict.
The LTTE did not comment on the military's latest casualty figures.
Tens of thousands of people have died since the conflict began in 1972.
About 332 separatist fighters and 11 troops have been killed so far in January alone, according to military figures.