The group, which employs hundreds of staff at a major sales and customer support centre in Glasgow, said in a stock market announcement that Bain was offering 280p per share, a 37 per cent premium on the closing share price on 10 August.
The board of Esure has indicated to its private equity suitor that it would be “minded to recommend” a firm offer.
In a brief statement, the insurer said: “The board of Esure notes the recent movement in its share price and confirms that, having received an unsolicited proposal from Bain Capital Private Equity and its affiliates, it is in the advanced stages of discussing a possible offer for the entire issued and to-be-issued share capital of the company by Bain.
“The board of Esure has indicated to Bain Capital that it would be minded to recommend a firm offer for Esure if made by Bain Capital at the price set out in the proposal.”
It added: “There is no certainty that an offer will be made for the company.”
Under UK takeover rules, Bain now has until early September to announce a firm intention to make an offer or walk away.
Shares in Esure were up by 31 per cent by lunchtime in London.
The insurance group was founded in 2000 by businessman Sir Peter Wood, who had previously launched the Direct Line insurance operation in the mid 1980s. He remains the chairman of Esure Group.
The firm provides insurance products to more than two million drivers, home-owners, pet owners and holidaymakers across the UK.