"At a meeting between the two companies on 13 December, Hardy's advisers confirmed that the Hardy board would only be prepared to recommend a price substantially in excess of the final proposal," Beazley said in a statement.
"As a result, Beazley today announces the withdrawal of its interest."
Beazey submitted its final offer proposal earlier this month. Hardy's would be valued at about £180m based on a 350 pence per share offer.
Beazley's 350 pence proposed offer represented a premium of around 24 per cent to Hardy's closing share price of 282 pence on 13 December.
Lloyd's of London insurers have been at the centre of takeover speculation as an unpromising trading environment has weighed on their share prices, opening up potentially attractive acquisition opportunities.
In October, Brit Insurance agreed to be taken over by buyout firms Apollo Management and CVC Capital Partners after a four-month takeover battle.