Beckman Coulter Inc., which makes diagnostic instruments used in clinical testing, has put itself on the block and could fetch more than $5 billion in a sale, people familiar with the matter said.

Several large private-equity firms and rival companies have expressed interest in Brea, Calif.-based Beckman Coulter, which had a market value of just under $4 billion Thursday, the people said. Goldman Sachs Group Inc. is running the sale process, which began a few ...