Ant, the finance affiliate of Alibaba, increased its offer to $18 per share in cash from $13.25, and the transaction is valued at around $1.2 billion, a statement by Ant and MoneyGram said.
Shares of MoneyGram closed up over 7 percent on Monday.
A successful deal would be the Ant's first major step to expand its business overseas. Ant, valued at around $60 billion, is also planning an initial public offering.
"MoneyGram... will add valuable cross-border remittance capabilities to the Ant Financial ecosystem, serving our more than 630 million users globally," Doug Feagin, President of Ant Financial International, said in the statement.
The two firms said they have made progress towards obtaining the regulatory approvals necessary to complete the transaction, including winning U.S. antitrust clearance.
The deal will also be reviewed by the Committee on Foreign Investment (CFIUS), a U.S. inter-agency panel that looks at foreign acquisitions for national security risks.
CFIUS has been a stumbling block for several Chinese deals in the United States and is considered a big hurdle for Ant Financial.
A Euronet deal is likely to be more agreeable to U.S. policymakers amid rising tensions between China and the United States over trade and foreign policy.