EasyJet has opened talks with Castlelake, despite rejecting a fourth takeover offer worth £4.9bn from the US investment firm, with the airline saying it would open its books in the hope of receiving a higher bid.

The British low-cost carrier unanimously rejected the latest proposal, of 650p a share, saying it still “substantially” undervalued the company while flagging “significant questions of deliverability”.

However, in a statement on Thursday, easyJet suggested a slight thawing in its positions.

The airline said: “The board believes that giving Castlelake access to limited commercial information, as Castlelake sought in the letter which contained the fourth proposal, might produce a more attractive proposal that better reflects the value of easyJet and its prospects and the interests of shareholders thereto.”

It added: “The board continues to be concerned about the ownership structure and deliverability of any offer from Castlelake, and the time it will take.”

Castlelake has until 5pm on 5 July to improve its offer or walk away, extended from a previous deadline of 26 June. The Minneapolis-based firm responded with a short statement, saying it welcomed the easyJet board’s “constructive engagement and the nine-day extension to the ‘put-up or shut-up’ deadline”.

The airline’s share price was up 8% on Thursday at 582p.

Castlelake’s latest proposal, of 650p a share, was higher than its previous 625p a share offer, described by easyJet as “highly opportunistic”, and up from 403p in its first proposal, made on 1 June.

The firm, which has $38bn (£28bn) in assets under management, is known for investing in aircraft leasing and aviation finance and already owns a small stake in easyJet.

To ​maintain an EU ‌operating licence, ​European carriers must be majority EU-owned and controlled.

Castlelake said it hoped to be able to further improve its bid after scrutinising easyJet’s books.

The US firm has added the New York-based asset manager Brookfield Asset Management to the bidding vehicle, alongside its two other partners, two Irish aviation executives. They are Peter Bellew, a former Malaysia Airlines chief executive and former chief operating officer at easyJet, Riyadh Air and Ryanair, and Mark Breen, the chief executive of Dublin-based Oneiros Aerospace and a former chief operating officer at Oman Air.

Under the previously ⁠proposed terms, the bidding vehicle would be owned 49% by Castlelake ​and co-investors including ​Brookfield. The remaining ​51% would be owned by Bellew and Breen, both EU nationals.

Bellew runs Dooks Capital, a seed investment and advisory firm focused on AI in aviation that he founded last September. It operates out of Saudi Arabia.

Chris Beauchamp, the chief market analyst at the investing platform IG, said: “EasyJet’s board might be making a decent showing of rejecting the Castlelake bid, the deadline extension has been taken as a sign that some kind of deal is doable. Investors clearly expect a sweetened deal to come through, which accounts for the continued strength in the shares over the past week.”