FMI Financial Corp., controlled by Cincinnati financier Carl Lindner, said yesterday it is offering $20 a share for up to 4 million shares of United Brands Co. common stock and reserves the right to buy up to 4.5 million shares. United, which closed yesterday at 19 1/2, up 1 1/2, is a major producer and distributor of food products.

If FMI acquires 4 million United shares, its stake in the company would increase to 33.7 percent from 7.3 percent. If it acquires 4.5 million shares, its ownership would increase to 37 percent. Lindner's overall stake in the company would increase to about 90 percent because another company he controls, American Financial Corp., has an additional 54 percent stake in United.

FMI's offer, which expires Aug. 9, is not conditioned on a minimum number of shares being tendered. Linder, who is chairman and chief executive officer of United, already controls the company's board and has appointed four of its nine directors. In the year ended March 31, United had revenue of $3.22 billion and net income of $23.1 million, compared with revenue of $3.4 billion and net income of $46.2 million in the prior year.