KKR and a partnership between TPG Capital and Onex Corp are also through to the bidding round alongside industrial machinery maker SPX Corp, which is pursuing a potential merger according to Reuters.
Blackstone and Bain Capital previously took part in the auction but did not make it through to the second round, the report said, in a sale that is expected to fetch about $3.5bn.
Gardner was founded in 1859 and makes industrial compressors, blowers, pumps and fuel systems.
The US-based business has 40 manufacturing facilities globally and posted revenues of $2.4bn last year, about two-thirds of which were generated abroad.
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