By Madeline Chambers and Paul Carrel

BERLIN (Reuters) - Angela Merkel's shock admission that she wishes she could turn back the clock on her migrant policy is a clear attempt to mend fences with her allies in Bavaria and a strong hint she will seek to run for a fourth term as German chancellor.

Not everyone in the conservative Bavarian CSU was convinced by her uncharacteristically contrite remarks on Monday, however, and Merkel's own views on political longevity could play into her decision on whether to stand again.

Merkel's conservative CDU suffered its second electoral rout in as many weeks in Berlin on Sunday as voters rejected her open-door migrant policy just a year before a federal election.

Striking a confessional tone that reached out to Catholics in her own party and in the CSU's southeastern heartland, a front line of the migrant crisis, Merkel took her share of the blame and said she would if she could "turn back time by many many years" to prepare Germany for the influx.

Looking more vulnerable than ever before in her 11 years in office, Merkel chose her words carefully as she sought to appease her CSU allies, who have repeatedly voiced anger at her decision to let in a million migrants last year.

The CSU accounts for roughly 20 percent of the conservative bloc of votes in the federal parliament and she needs its support to stand again for the chancellorship.

Reading a script, she distanced herself from her "we will manage it" mantra on integrating refugees which has been widely criticized by political foes in the CSU as well as the anti-immigrant Alternative for Germany (AfD), which is winning votes.

But, crucially, she offered no policy concessions, which means that this may not go far enough for the CSU to endorse her in the next few months.

"She is weaseling without revoking," said Josef Joffe, publisher-editor of Germany weekly Die Zeit, adding the underlying message was 'I was not wrong. I only failed to get the message across'.

TIME TO GO?

Inscrutable Merkel smiled and declined to comment when asked if she would run in 2017, saying only she was still motivated.

Spiegel Online and top-selling daily Bild clearly interpreted her remarks as a signal that she would stand again.

"There can be no doubt over whether she will run again," Bild wrote in an editorial, reflecting a widely held view in Berlin that there is no obvious successor.

However, there was some dissent within German media. The Berlin correspondent of Deutschlandfunk radio said he would not be surprised if she smoothed things over with the CSU in the next couple of months only to announce at the CDU's conference in December that she will not run again.

Merkel, 62, who grew up in Communist East Germany, is well aware that leaving politics at the right time is one of the most difficult decisions a leader has to make.

"I would like some day to find the right time to leave politics. That is much harder than I had previously imagined it to be. But I don't want to be a half dead wreck when I leave politics," she was quoted as saying in 1998.

Merkel was herself ruthless towards her mentor, former Chancellor Helmut Kohl, saying in the midst of a party funding scandal in 1999 that it was time to move on without him.

"Of course she will run again. It would be irresponsible for her to throw in the towel now," one senior German official said. "She will do it again.. I would bet money on this."

However, even the fact that there is speculation about her future highlights how things have changed. Until only a few weeks ago, the question was not even raised.

"Instability is bad and all investors love Merkel, but one is reminded of when (former British Prime Minister Margaret) Thatcher became very beloved by foreigners and foreign investors and started losing luster back home," said Sassan Ghahramani, CEO of U.S.-based SGH Macro Advisors which advises hedge funds.

EYES ON BAVARIA

Much depends on whether Merkel's contrition will win over her critics in the CSU, focused more squarely on their party's performance on its Bavarian home turf than on national politics.

Senior CSU lawmaker Stephan Mayer told broadcaster NDR that Merkel's comments were a "very important and welcome signal" showing she had understood popular concerns, adding the two parties were on the path of reconciliation.

But CSU boss Horst Seehofer was lukewarm, saying voters did not want backward-looking apologies, but actions.

The influx of migrants has already slowed from the million who entered Germany last year and is estimated to be nearer 300,000 this year. Even last December, Merkel acknowledged her critics' fears and said the flow of refugees must ease.

That has been achieved, partly because other countries have closed the Balkan route for migrants. Merkel argues other measures, including an EU-deal with Turkey, have also helped.

A humbled Merkel may have changed her vocabulary, but her statement on Monday was also an implicit rejection of the central CSU demand of a cap of 200,000 migrants a year.

Former Interior Minister Hans-Peter Friedrich, of the CSU, told Focus Merkel realized she had to adjust her message but as long as she rejected a cap, doubt remained over her resolve.

"So much trust has been lost, you also have to change policies," he said.

(Additional reporting by Noah Barkin and Andreas Rinke; writing by Madeline Chambers; editing by Philippa Fletcher)