NAIROBI (Reuters) - Kenya's opposition on Thursday softened its rejection of a presidential initiative to end a dispute over an electoral oversight body, suggesting the move was a start in accepting the need for dialogue but did not go far enough.

The opposition has been staging almost weekly protests against the Independent Electoral and Boundaries Commission (IEBC), which its says is too biased to oversee a fair vote in 2017.

The demonstrations have led to regular clashes with police in Nairobi and at least four deaths in opposition strongholds in western Kenya. The violence in a nation prone to political strife has alarmed Kenyans and Western diplomats.

In a bid to resolve the dispute, President Uhuru Kenyatta proposed on Wednesday the setting up of a bipartisan committee in parliament to discuss the issue, saying this would meet the demand for dialogue as well as adhere to constitutional rules.

After rejecting the idea as "nothing new" on Wednesday, opposition leader Raila Odinga's spokesman, Dennis Onyango, moderated his position after talks on Thursday between the opposition and church leaders, who are acting as mediators.

"We are now saying at least he has recognized the need to discuss IEBC but that alone is not enough," he told Reuters in a mobile phone text message. "If our concerns are not taken on board we will resume protests."

He did not specify the concerns but said they would be relayed by mediators to the government.

The opposition has in the past dismissed the idea of seeking changes to the IEBC via parliament, a body dominated by Kenyatta's Jubilee coalition and which previously rejected an opposition petition against the IEBC.

The IEBC has denied charges of bias and its commissioners have said they would not quit. The opposition criticized the commission after the 2013 vote, when it fumbled technology to identify voters and transmit results from polling stations.

A court rejected the opposition petition disputing the result.

Kenyatta met Jubilee lawmakers to draw up a list of names from his coalition for a bipartisan committee.

"We are ready to engage in the matter of the IEBC. The engagement will happen through a Joint Select Committee of Parliament," the Jubilee parliamentary groups said in a statement released by the president's office.

(Writing by Edmund Blair; Editing by Gareth Jones)