(Recasts with changed sourcing; adds Hellman & Friedman declined to comment in paragraph 8, background)

June 23 (Reuters) - Software company Zendesk Inc, which has a market value of over $7 billion, is close to a deal with a group of buyout firms that includes Hellman & Friedman LLC and Permira, people familiar with the matter said on Thursday.

A deal could be struck in the coming days, they said. Zendesk's shares rose 41.5% to $81.98 after the bell.

The San Francisco-based firm, which has been under pressure from activist investor Jana Partners, said earlier this month it would continue as an independent public company after completing a review of its business.

Jana had nominated four directors to the board of Zendesk, arguing the company must be rehabilitated after an unpopular attempt to buy Momentive Global, parent of web-survey firm SurveyMonkey.

The Wall Street Journal had reported last week that Zendesk was in talks to settle with Jana, which could have involved Mikkel Svane stepping down as its chief executive and some changes to the board.

However, the newspaper, which first reported the buyout talks on Thursday, said that it wasn't clear where Zendesk's discussions with Jana stood with this expected deal with private-equity firms.

Zendesk, a customer service software vendor, has seen a surge in business during the pandemic, as its enterprise customers accelerate digital transformation.

Hellman & Friedman and Permira declined to comment on the buyout talks, while Zendesk did not immediately respond to Reuters' request for comment. (Reporting by Akash Sriram and Mrinmay Dey in Bengaluru and Chibuike Oguh in New York; Editing by Rashmi Aich)