Lloyd's of London insurer Beazley said smaller peer Hardy Underwriting had rejected a takeover approach made earlier this month but that it still hoped to clinch a deal.
Dublin-based Beazley said that on 6 October it had proposed paying 300 pence per share in cash for Hardy, valuing the company at about £155m.
The approach, which represented a 36 per cent premium to Hardy's closing share price on 5 October, was rejected by the Bermuda-based insurer in a letter dated 8 October.
Consolidation amongst the Lloyd's insurers, which offer cover against large-scale risks such as natural disasters, has long been mooted, especially among the smaller players, but deals are difficult to secure.
Beazley, which in July reported that its first half profit nearly quadrupled, said it will continue talks with Hardy's board and shareholders "with the intention of agreeing a recommended transaction".
"Beazley believes that the two groups are highly compatible. Both have well-regarded underwriting teams, renowned track records of profitability and a shared objective to develop a diverse speciality insurance franchise," the insurer said in a statement.