Starace, 67, has held the job since 2014 but his mandate expires in May and the sources, who asked not to be named due to the sensitivity of the matter, have said a fourth term looks unlikely.
Stefano Donnarumma, currently chief executive of power grid Terna, is among those shortlisted to replace Starace along with Flavio Cattaneo, deputy chairman of Italian high-speed rail firm NTV and former head of phone group Telecom Italia (TIM).
Donnarumma and Cattaneo were not immediately available for comment. Enel declined to comment.
Discussions are still ongoing within the cabinet and will enter a crunch phase after important regional elections in Lombardy and Lazio in mid-February, the sources said, the first electoral test for Meloni's three-month-old government.
Meloni made it clear she was looking to put her stamp on key jobs this month, when the government appointed veteran economist Riccardo Barbieri as Treasury director general, ousting Alessandro Rivera.
Besides Enel, seats on the boards of other major state-controlled companies will be up for grabs in the coming weeks, including energy group Eni, bailed out bank MPS and defence group Leonardo.
TOO MUCH DEBT?
Starace in 2020 was reappointed for a third term, after winning plaudits from investors for his financial results and for transforming the former power monopoly into one of the biggest and greenest utilities in the world.
Three years on, Meloni's office sees Enel's growing net debt, which is expected to be around 60 billion euros ($64.94 billion) at the end of 2022, as a worry, one of the sources told Reuters.
Hit by soaring gas prices and government measures capping bills to shield consumers across the world, Enel's net debt is estimated to be 3.1 times its core earnings at end-2022. This compares with a 2.5 ratio at the end of 2019.
Meloni's top tier also considers Starace as too independent, the sources said.
In early 2022, Starace rebuffed calls from the office of former Prime Minister Mario Draghi to skip a business video conference between Russian President Vladimir Putin and the heads of some of Italy's top companies, just a month before Moscow's invasion of Ukraine.
Further back, government representatives under Draghi's predecessor, Giuseppe Conte, accused Starace of dragging his feet over a plan to merge Enel-backed broadband operator Open Fiber with TIM's landline grid.
Enel in 2021 sold its 50% stake in Open Fiber to state lender Cassa Depositi e Prestiti (CDP) and Australian fund Macquarie.
($1 = 0.9240 euros)
(Additional reporting by Francesca Landini and Elvira Pollina in Milan; Editing by Alexandra Hudson)
By Giuseppe Fonte, Angelo Amante and Gavin Jones