The shares and options of Sears Holdings Corp rose on Tuesday on speculation the operator of Sears department stores and the Kmart discount chain could go private.

There is a rumor that Bruce Berkowitz of Fairholme and Eddie Lampert could take Sears private, Jon Najarian, a co-founder of online stock, options and futures brokerage TradeMonster.com in Chicago said.

That rumor followed some unusual buying in Sears call options during the first 25 minutes of trading, he said. Buying of Sears stock and options moved up as the rumor of a leveraged buyout spread, he said.

Sears spokesman Chris Brathwaite said the company had no comment on market rumors or speculation.

The speculation comes just days after a major business lender, CIT Group Inc , halted loans that Sears' suppliers use to finance the goods they sell to the struggling chain.

Morningstar analyst Paul Swinand also said the shares were up on a Lampert writes a check scenario.

It would make sense to be private, but they still have to improve the operating structure and costs, pension, etc., then reinvest in stores, and then get appliances to sell again, Swinand added.

Sears shares were trading at double their full session average by 11:30 am EST. Its option volume was 2.9 times the average daily levels, with 44,000 calls and 16,000 puts traded during the first two hours of the session, according to options analytics firm Trade Alert.

According to documents filed last Wednesday, Lampert, who owns directly and through related entities about 59 percent of the company, bought roughly $159 million worth of Sears shares from his ESL Investments hedge fund, representing 4.46 million shares in private sales. In addition, Lampert also bought shares on the open market for about $12 million.

Many were skeptical about Sears' chances of going private.

We think the company has bigger issues that could make a going private transaction impossible, including significant cash burn with rapidly deteriorating financial performance and reduced liquidity, particularly if the major vendors become reluctant to fund holiday 2012 shipments, Mary Ross Gilbert, managing director, Imperial Capital said.

Sears shares were up about 9.3 percent at $36.67 in midday trading on the Nasdaq. They touched a high of $38.80 earlier in the session.

(Reporting By Dhanya Skariachan and Doris Frankel; editing by Andre Grenon)