Conrail Chairman L. Stanley Crane has formally submitted a management proposal to sell Conrail through a public stock offering, although there is evidence that Norfolk Southern Corp. would be the government's preferred buyer if its offer can pass Justice Department review.

Crane's management proposal, sent with the concurrence, but not the endorsement, of his board of directors, offers $1.4 billion in cash for the Northeast/Midwest freight railroad. That is about $200 million more than the three bidders the Transportation Department has selected as finalists in its attempt to denationalize Conrail.

Those bidders are Norfolk Southern, a major East Coast railroad holding company; Alleghany Corp., a New York holding company with a long history of railroad ownership; and a group of investors headed by hotelier J. W. (Bill) Marriott Jr.

Transportation Secretary Elizabeth Hanford Dole sent Norfolk Southern's offer to the Justice Department for review in October to address potential antitrust questions. There is some overlapping of Conrail and Norfolk Southern routes west of Pittsburgh, and Conrail already has a virtual monopoly on rail access to the New York metropolitan area.

The transportation section of Justice's antitrust division has been interviewing shippers, checking routes and revenue, and asking questions of Norfolk Southern and Transportation Department officials since then. "Our focus is competition and consumer welfare," a Justice official said. He said that Justice is close to completing its work.

It sounds much like the kind of negotiation Justice would conduct with a private company prior to stating it would not oppose a merger if certain conditions were met. Norfolk Southern officials are known to feel that they can meet all the conditions Justice has suggested so far and that, if they do, they will be DOT's recommended bidder.

A source said "there is no question that, if Norfolk Southern is cleared, that would be the strongest possible offer." Dole's first criteria always has been to select a buyer that "leaves Conrail in the strongest financial position after the sale."

Although Norfolk Southern is clearly the strongest financial bidder, the question that remains in the minds of Dole and her advisers is whether it would remain in that condition after it meets all of Justice's objections and concludes unfinished negotiations with Conrail's employes. "You can't make the final judgment until you get those things tied down," an official said.

Dole has been placed on notice by CSX Corp., Norfolk Southern's major rival in the Southeast and Midwest, that it will fight a Norfolk Southern takeover of Conrail. CSX Chairman Hays T. Watkins said in a recent letter to Dole that such an event "strikes at the vital interests of CSX and will be resisted by every resource at our command and in every forum where the challenge can be brought."

Watkins reiterated that in a meeting with Dole. "We just wanted to make sure there was no misunderstanding," a CSX source said. "We heard them," a Transportation Department source said.

There are no obvious antitrust questions in the other two offers, and Justice is not reviewing them. O. M. Berge, chairman of the Railway Labor Executives Association, said yesterday "we are very close in principal with Alleghany and Marriott; we have not had in-depth discussions with Norfolk Southern."

Conrail's employes also have bid for the railroad, although their effort has been seen from the beginning as an attempt to ensure that they would have a major role when the bidding reached the final stage.

Rep. James J. Florio (D-N.J.), chairman of the House subcommittee that must approve any Conrail sale, wrote Dole yesterday and told her Crane's offer "merits serious consideration." Crane forwarded his proposal to Dole after a Conrail board meeting last Friday. He has long favored a public offering of Conrail stock, something Dole and her advisers feel has more drawbacks than a straight sale to a strong company.

Under Crane's plan, Conrail would raise $500 million in a common stock offering and place $600 million privately in preferred stock. Another $300 million would come from Conrail's cash reserves. That would make it one of the largest public offerings in history.

Crane's plan also assumes that employe ownership of Conrail would increase from its present 15 percent to 30 percent and that Conrail employes' wages would be restored to industry levels effective last July 1. Employes have been accepting reduced wages.