In a letter to two top executives at Dollar Thrifty, Avis' chairman and CEO said he wants Dollar Thrifty to open its books before his company formally counters the $1.17 billion offer from Hertz Global Holdings Inc.
That sent shares of Dollar Thrifty, based in Tulsa, Oklahoma, to a three-year high.
"We would like to make a substantially higher offer to acquire Dollar Thrifty, especially in light of your recent performance," CEO and Chairman Ronald Nelson wrote.
Last week, Dollar Thrifty posted a record first-quarter profit, earning $27.3 million, even though revenue slipped almost 4 percent.
Monday's move comes a week after Hertz, the world's largest car rental company by locations, offered a cash-and-stock bid for Dollar Thifty that values the company $41 per share.
Regardless of who ultimately acquires Dollar Thrifty, the expanded company will be forced to compete in an increasingly consolidated rental car market.
Hertz, based in Park Ridge, New Jersey, owns the Advantage rental car company and competes against Enterprise Holdings Inc., which has 7,600 Enterprise, Alamo and National sites. Enterprise raked in $12.9 billion in revenue last year. Avis has 6,900 Avis and Budget locations and about $5.13 billion in revenue
Nelson said executives at his company have been interested in a possible acquisition of Dollar Thrifty for "several months," going as far as to try to set up meetings with leaders to talk about a sale, including a dinner planned for last week that was canceled after the deal with Hertz was announced.
"It is hard to understand how your failure to engage in discussions with an interested strategic buyer, who you know also would be able to achieve significant synergies as a result of a combination, can be consistent with the fiduciary duties that you and your board carry to seek the best possible deal for your shareholders," he wrote.
Officials at Dollar Thrifty and Hertz did not immediately return messages seeking comment.
Dollar Thrifty shares soared as high as $50.68 Monday morning and were up $6.25, or 14.2 percent, to $50.24 by mid morning.
Avis Budget, based in Parsippany, New Jersey, fell 14 cents to $14.98, while Hertz tumbled 53 cents, or 3.7 percent, to $13.93.
By AP Retail Writer Ashley M. Heher; AP Transportation Writer Samantha Bomkamp contributed to this report from New York