Blackstone Infrastructure's latest offer of $22.45 per Class A share represented a premium of 22.7% to the company's last close on Monday.
Blackstone Infrastructure Partners along with affiliates Spain's Enagas, GIC, NPS, and USS already own nearly 44% of the total Class A and Class B shares of Tallgrass.
The U.S. energy firm had initially received a take-private offer from Blackstone and its partners in August, following which Tallgrass had commissioned a Conflicts Committee.
The committee has unanimously approved the transaction, the statement said on Tuesday.
The transaction will value the company's public float at about $4.02 billion, based on the number of Class A shares of 179.2 million, as of October-end.
Blackstone Infrastructure will fund the transaction with about $3 billion in equity and remaining by debt, Tallgrass said.
In a separate statement, Spanish firm Enagas said it has agreed to invest $836 million in the U.S. pipeline operator as part of a takeover involving other investors.
Citigroup Global Markets Inc and Credit Suisse Securities are the financial advisers to Blackstone Infrastructure Partners, while Evercore Group acted as the financial adviser to Tallgrass.