The price is 3.2p more per share than the top end of the 50p to 55p range previously suggested by Apax, and values Woolies at approximately£837 million.
Royal Bank of Scotland is lined up to provide Apax with a debt package to help fund the acquisition.
The Takeover Panel had told Apax that it must make its intentions on Woolworths clear by midday on Monday, but a new May 6 deadline for a formal offer has now been set.
Broker Numis believed 58p was a full price, but it was still far from certain that a deal would happen. 'Due diligence has not yet started and all VCs are well known for 'chipping' initial indicative values,' the broker said.