Shares closed Tuesday slightly below $16, up more than 24 percent for the day.
Euronet's offer values the company at more than $1 billion, which is 15 percent above a previous offer that was made by a U.S.-based money transfer company and Alibaba's Ant Financial Services.
Just earlier this year, shares of MoneyGram jumped after Ant Financial proposed its merger in a deal valued at $880 million. In that deal, shareholders of Dallas-based MoneyGram were offered $13.25 per share in cash, according to a press release.
By combining with Ant Financial, MoneyGram would be able to access 630 million more users worldwide, the release said.
The latest Euronet proposal "presents a faster and a significantly more certain path to closing than the Ant Financial transaction," Euronet Chief Executive Michael Brown wrote in a note to MoneyGram Chairman Pamela Patsley on Tuesday.
"The combination of these highly complementary businesses would create a very well-positioned global payments company that would benefit customers and employees in the United States and around the world," Tuesday's press release from Euronet wrote.
MoneyGram did not immediately respond to CNBC's request for comment on this proposal.
Shares of Euronet traded down more than 1 percent on Tuesday's news but later closed the day up a little more than half percent.
— CNBC's Natalia Wojcik contributed to this report.