The company could fetch more than €500m according to Reuters, which cited two people familiar with the matter.
A third source added that EQT had hired Swedish law firm Vinge as legal advisor, Reuters added.
The investment was the third made by the EQT Infrastructure fund, and the first investment in Sweden for the firm.
Swedegas owns and operates the gas grid in Sweden, which extends from Dragör in Denmark to Stenungsund in Sweden.
The grid supplies natural gas to 33 municipal areas and several combined heat and power plants. Natural gas is also used in 37,000 households and in the transport sector.
Swedegas also plans to transport biogas in the grid in the near future.
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