The United Transportation Union, the largest labor organization at Conrail, yesterday proposed a new plan for the public sale of the federal government's share of Conrail that would be underwritten by other railroads but keep Conrail independent.

The UTU represents 8,000 of Conrail's 35,000 employes. Its plan is but one of many ideas kicking around now that Rep. John D. Dingell (D-Mich.), chairman of the House Energy and Commerce Committee, has come out against Transportation Secretary Elizabeth Hanford Dole's proposal to sell Conrail for $1.2 billion to Norfolk Southern Corp., the Norfolk-based railroad holding company. The Senate has approved that proposal.

Dole called Monday for Norfolk Southern to increase the price to $1.9 billion to deal with one of Dingell's objections, what he considered a low sales price, and Norfolk Southern Chairman Robert B. Claytor said yesterday that "I have advised the secretary we will consider" her suggestion.

Dingell, however, said yesterday that "the proposed sale of Conrail to Norfolk Southern remains unacceptable and . . . my position is final. It is time to move forward to put together a public offering."

Dingell's opposition appears to be based on anticompetitive and labor protection issues. Norfolk Southern and Conrail serve many of the same Midwestern markets and would be required under any merger plan to divest some of their shared holdings to other railroads.

The UTU plan, which union president Fred A. Hardin outlined yesterday to key congressional and administration officials, calls for a public sale of the federal government's 85 percent holding in Conrail to raise $1.6 billion. Conrail's cash on hand, now more than $900 million, could be reduced by $300 million to make a total of $1.9 billion, matching the figure Dole is asking of Norfolk Southern, Hardin said.

Unsold stock would be purchased by "several other railroads who have indicated a sincere interest in such a proposal," Hardin said. He declined to identify those railroads, but said major roads are interested. The railroads purchasing Conrail stock "would have no vote or voice in the operation of Conrail but would know that it would remain a strong and efficient carrier, to complement their rail services throughout the country," Hardin said.

Should Conrail fall on the financial hard times Dole says are inevitable, those railroads would be required to purchase additional stock to keep Conrail well.

Hardin is attempting to address two of Dole's problems with a public offering: the long-term viability of Conrail, which she says is questionable despite its current impressive financial performance, and the threat that if the stock market goes bad while Conrail is being sold, the federal government will become a minority stockholder with all the responsibility and none of the authority.

In a letter to Dingell on Monday, Dole anticipated the Hardin proposal and said it "raises many questions. For example, what conditions would relieve the underwriters or the guarantors from meeting their minimum purchase price obligations to the United States?"

Although Dingell has called for a public offering and an independent Conrail, he has not endorsed either existing public offering plan, one from Morgan Stanley & Co., the other from Conrail Acquisition Corp., a joint effort of Allen & Co. and First Boston Corp.