* U.S.-China trade hopes underpin market
* Trade turnover low as investors still half-hearted
* Japan Post Insurance jumps on further privatisation
* Recovery in Lyft boosts Rakuten
* Foreigners net sellers of Japanese stocks March 25-29 - TSE
By Ayai Tomisawa and Hideyuki Sano
TOKYO, April 5 (Reuters) - Japan's Nikkei edged up on Friday to a fresh one-month high as expectations of an imminent U.S.-China trade deal prompted investors to buy back shares they had sold, while Japan Post Insurance, or Kampo, jumped on hopes of deeper privatisation.
The Nikkei share average rose 0.38 percent to 21,807.50, the highest close since March 4. For the week, the benchmark index was up 2.83 percent, the biggest weekly gain in about three months.
China's Xinhua news agency, quoting Vice Premier Liu He, reported that President Xi Jinping said progress was being made and called for an early conclusion of negotiations.
U.S. President Donald Trump said that trade talks with China were going well and he would only accept a "great" deal as negotiators hammered out differences ahead of a meeting between Trump and China's vice premier on Thursday.
Shoji Hirakawa, chief global strategist at Tokai Tokyo Research Institute, said investors "have already taken heart from signs of progress in the talks, and if a deal comes out, it could be another booster for the stock market".
Investors are also waiting on the U.S. payrolls report which is forecast to bounce back by 180,000 in March, following February's distorted 20,000 rise.
Data from the U.S. Labor Department showed that jobless claims fell to a 49-year low last week, pointing to sustained labour market strength.
Japan Post Insurance soared 9.9 percent after an announcement the previous day that its parent Japan Post Holdings is selling nearly a third of its stake, a step aimed at advancing the privatisation of the Japan Post group.
The plan was seen as giving the insurer larger management freedom while its announcement of a share buyback of up to 50 million shares for as much as 100 billion yen ($895.4 million) also gave an additional boost.
Rakuten gained 6.6 percent as shares of Lyft , of which it is the largest shareholder, recovered to their IPO price.
The broader Topix rose 0.35 percent to 1,625.75. Still, trade turnover was low at 1.95 trillion yen, about 20 percent below the average, suggesting investors' tepid conviction in the rally.
Indeed, foreigners were net sellers of Japanese stocks for the second consecutive week ended March 29 as a fall in U.S. bond yields stoked fears of a recession in the United States.
Overseas investors sold a net 857.5 billion yen of Japanese stocks, including cash equities and futures in that week, marking their biggest weekly net sales since Oct. 26, 2018, data from Japanese stock exchanges showed.
$1 = 111.68 yen Editing by Jacqueline Wong