The firm has offered $3 billion in the deal, around eight per cent per share above the US lender's closing price yesterday.
Overall, UnionBanCal is valued at $8.8 billion under the terms of the offer.
Tokyo-based Mitsubishi has been hit by significant headwinds at home, with profits growth on the decline in Japan.
Therefore, it is now looking to expand its overseas holdings, with UnionBanCal a prime target.
A lower offer was previously rejected by the San Francisco-based bank - which retains 330 branches across the US - in April.
Commenting on the new bid to Bloomberg, Nomura analyst Keisuke Moriyama said: "UnionBanCal's earnings are not so bad, and Mitsubishi has decided to take full benefit of the bank's revenues.
"California is a growth area and it will be easier for Mitsubishi to manage than buying a bank from scratch."