IC Industries, the Chicago-based conglomerate which controls Illinois Central Gulf Railroad, has proposed a $390 million merger with Pet Inc.

In an announcement here yesterday, Pet said it had received an unsolicited letter from IC proposing a merger under which Pet shareholderes would receive $54 in cash for each share of Pet common stock they own.

IC also proposed an alternative plan under which 45 percent of Pet's 6.6 million outstanding shares would be converted into cash with the remainder converted into an equity security of IC Industries.

Pet, which closed at 42 7/8 Friday, closed up 9 1/2 points yesterday at 52 3/8. IC Industries closed up 5/8 at 25 3/8.

A company spokesman for Pet said the St. Louis-based corporation is consulting with its investment bankers, Morgan, Stanley & Co. and said its board of directors would meet this weekend to discuss the merger proposal.

Pet's management cautioned shareholders not to act hastily and to wait for Pet management's review of the offer.

IC Industries also controls Midas Mufflers, Pepsi-Coca General Bottler, Abex Corp. (a manufacturing company) and has several other interests. IC had sales last year of $1.87 billion and earnings of $78.5 million ($4.55 a share).

Pet, best known as a dairy company but also with wide interests including Whitman's Candies, had $1.1 billion in sales last year.

According to Pet, the IC letter said that Pet would continue to operate as a separate company.