* Sterling crawls higher, down more than 7 pct in June on Brexit

* Dollar slips more than 7 pct in June vs perceived safe-haven yen

By Lisa Twaronite

TOKYO, June 30 (Reuters) - The dollar took a breather in Asia on Thursday but remained near a 3-1/2-month high against a basket of currencies hit in the wake of Britain's stunning vote to exit from the European Union, while recently battered sterling crawled higher.

The dollar index, which tracks the greenback against a basket of six major rivals, edged down 0.1 percent to 95.642 , on track for a monthly loss of 0.2 percent.

It rose as high as 96.705 on Monday, when the pound plumbed 31-year lows after the results of the UK vote on Friday sent markets reeling.

Sterling was up 0.3 percent at $1.3456, well above Monday's nadir of $1.3122 but still poised to lose more than 7 percent for the month.

The euro was down 0.2 percent at 82.67 pence edging away from Monday's high of 83.80, which was its loftiest peak in over than two years.

"The odds may be against them but investors are hoping that the worse is over for currencies and equities and the gaps on Friday will be filled," Kathy Lien, managing director of foreign exchange strategy at BK Asset Management, wrote in a note to clients.

"But considering there's been had no additional clarity on the terms of Brexit or the outlook for the UK economy and global economy since Britain's decision to leave the European Union on Friday, we don't see fundamental support for the recent moves," she said.

Chaos also continued in Britain's political arena, as both main opposition Labour Party and ruling Conservative Party headed for leadership battles.

The euro remained under pressure, down 0.1 percent at $1.1116, but remained well above its 3 1/2-month low of $1.0912 hit on Friday last week and down only 0.1 percent for the month.

Against its perceived safe-haven Japanese counterpart, the dollar edged down 0.1 percent to 102.72 yen, well above its 2-1/2-year low of 99.00 hit in volatile trade on Friday.

It was still on track to shed more than 7 percent for the month.

The yen tends to appreciate sharply in times of global risk, creating headaches for the Japanese government and threatening to push the country back into recession.

Data released earlier in the session showed that Japan's industrial output fell in May at the fastest rate in three months, highlighting concerns about falling exports and doubts about weak consumer spending.

(Reporting by Lisa Twaronite; Editing by Kim Coghill)