Endurance's cash-and-stock offer of $47.50 per share represents a premium of about 21 percent to Aspen's Friday close on the New York Stock Exchange.
"Despite our repeated attempts since late January to engage in confidential and friendly discussions, Aspen's board and management have rebuffed our proposal and refused to engage with us," Endurance Chief Executive John Chairman said in a statement.
Aspen's shares were trading well below the offer price at $43.69 before noon. Endurance shares were down 1.5 percent at $53.
"With Aspen's half of the premium coming from casualty line ... there is more reserve risk then some of the other deals we have seen in past couple of years," Keefe, Bruyette & Woods Inc. analyst Meyer Shields said, adding that he was skeptical of most insurance companies' acquisition activity.
Aspen said the proposal undervalued the company, represented a strategic mismatch and carried significant execution risk.
"Endurance has shown a public disdain for Lloyd's [of London insurance market], which is the growth engine of Aspen's well-established international insurance business," Aspen said in a statement.
Endurance, however, said its crop insurance products would complement Aspen's businesses, including its Lloyd's operations.
Endurance also estimated annual savings of more than $100 million and said it expected a potential deal to add to earnings in 2015.
The company said it would fund the acquisition partly through proceeds of $1.05 billion from placement of new shares to investors led by CVC Capital Partners Advisory and its affiliates.
Morgan Stanley & Co. and Jefferies LLC are acting as financial advisers to Endurance, while Goldman Sachs is advising Aspen.