Musgrave has made a £231.6m (€360m) offer for Budgens.
It's offering 135p per share for the Middlesex-based supermarket firm.
Irish supermarket group Musgrave currently owns 28% of Budgens.
Musgrave's managing director, Seamus Scally, says: "We have come to know Budgens and its management well since the acquisition of our stake in August 2000.
"We are now satisfied that Budgens represents the right base from which to develop our position in the grocery retailing market in Great Britain."
Budgens says the offer is good value for its shareholders.