The Direct Line chief executive, Adam Winslow, has appealed to shareholders to give his team more time to turn around the struggling insurer, faced with an unsolicited £3.3bn offer from Aviva, while its bigger rival is trying hard to drum up support from investors for the takeover.

The two companies – the UK’s biggest insurer, Aviva, and the Churchill owner, Direct Line, known for motor cover and its red phone on wheels mascot – are facing off in a takeover tussle that has sent the Direct Line share price soaring, amid speculation that Aviva could raise its offer or launch a hostile bid, or face a counterbid.

Winslow, who took the helm at Direct Line in March, joining from Aviva, claimed the new management team was “making excellent progress in the early stages of a significant turnaround”. He told the Sunday Times that Direct Line had “great brands” and was better off pushing ahead on its own with a “a strong standalone story and clear path to create shareholder value”.

Direct Line last week rejected a non-binding proposal from Aviva worth 250p a share, made up of cash and Aviva shares, saying it was “highly opportunistic and substantially undervalued the company”, and has declined to engage further with the Aviva board. Direct Line’s chair, Danuta Gray, met the Aviva chair, George Culmer, last week to explain why the company could not accept a proposal that low.

Aviva, run by Dame Amanda Blanc, has been on a charm offensive, talking to a number of Direct Line shareholders to persuade them of the merits of its approach, and to get them to encourage the board to engage in talks. Several are also investors in Aviva, including Schroders, Fidelity, Redwheel and M&G.

Shares in FTSE 250-listed Direct Line jumped by 41% on Thursday after news of Aviva’s approach, and closed a further 4.6% higher on Friday at 234.8p, up 10.4p.

Aviva needs to raise its offer by “several hundred million pounds”, according to Sir Peter Wood – the Royal Bank of Scotland executive who founded Direct Line in Croydon in 1985 and revolutionised car insurance with the simple notion of cutting out the middleman, the broker, and selling direct, by phone. Wood told the Mail on Sunday that Direct Line had lost its way, and that it would take Winslow three to four years to turn the company around.

Analysts said there was scope for Aviva to raise its offer to 270p a share by increasing the share component of the deal, with some talking of 300p a share.

It is the second time this year that Direct Line has faced an unwanted suitor. The Kent-based company, which owns the Churchill, Green Flag and Darwin brands, rejected two takeover proposals from the Belgian insurance group Ageas in February and March, valuing it at £3.2bn.

Analysts said there could be a counterbid from Ageas. It is one of the largest car insurers in the UK and has more than 4 million customers across car, home, travel and business insurance. Its chief executive, Hans de Cuyper, said in October that Ageas was open to making further bids for UK companies.

Winslow’s management team includes other ex-Aviva colleagues, Jane Poole, now Direct Line’s chief financial officer, and Maz Bown, its chief risk officer. Winslow has announced 550 job cuts as part of a £100m cost savings programme after a smaller-than-expected return to profit in September.

The proposed takeover could raise competition concerns and mean higher prices for customers, according to James Daley, the managing director of the consumer group Fairer Finance.

Aviva has indicated its market share in UK personal car insurance is 8% and 12% in home insurance, where it is the market leader. Alex Mackenzie, an analyst at BNP Paribas Exane, estimates that Direct Line has a 12% share of the motor market, behind the market leader Admiral, and 10% in home insurance, “and so we expect that anti-trust constraints will be limited”.