More than 100 private colleges in eight states are expected to join Dartmouth College this summer in selling hundreds of millions of dollars in tax-exempt bonds to finance loans to students made ineligible for federal help by budget cuts and eligibility restrictions.

Dartmouth announced yesterday that it had raised $29 million through the sale of tax-exempt bonds. The Hanover, N.H., college plans to use $12 million to make loans to needy students rendered ineligible for loans by cuts in the federal student aid programs. It says it will use the rest of the $29 million to finance campus construction projects.

Dartmouth is the first private college to raise funds in the financial markets for student loans. The move was made in direct response to threatened cutbacks in federal financial help for college students, Dartmouth officials said.

The Reagan administration proposed in February that the federal government's $6 billion program of aid to college students be cut by $900 million. Most existing aid programs, however, should remain intact this year, according to Robert Aaron, public affairs director for the American Council on Education.

Financial aid officers are worried about further cuts in federal aid next year, Aaron said, adding they are also concerned about the new tightened eligibility requirements for the Guaranteed Student Loan program.

The irony for the administration is that this increasing use of tax-exempt bonds represents an additional loss to the Treasury, which does not collect taxes on the interest received by bondholders.

Dartmouth officials said it was primarily the uncertainty over future federal aid that prompted their unprecedented move to the bond market.

Officials at Harvard University in Cambridge, Mass., said they also planned to raise several million dollars for student loans this summer in a joint effort with a number of other major private schools in the state. That offering would be made under the auspices of the newly created Massachusetts College Student Loan Authority.

If this summer's offering is successful, Harvard plans to return to the tax-exempt bond market for more money next year, according to Robert Scott, Harvard's financial systems director.

Scott said Harvard would use the low-cost money to "fill in around the federal program."

A representative of Northwestern University in Evanston, Ill., said it will seek $10 million to $15 million in the next two or three months to finance student loans next year. Plans are for it to join a dozen other private Illinois colleges in a $30 million to $40 million offering under the auspices of the Illinois Independent Higher Education Loan Authority.

Colby College in Waterville, Maine, is also planning a much smaller entry into the bond market this fall with similar state-authorized support.

At present, at least five states--New Hampshire, Massachusetts, Maine, Illinois and Iowa--have set up authorities that permit private colleges to offer tax-exempt bonds. At least four other states--New York, Maryland, Pennsylvania and Florida--have legislation in the works that would establish such authorities.

According to educational lobbyists here, more than 100, and maybe as many as 200, private colleges in the country may soon seriously consider offering tax-exempt bonds to get around cuts in federal student aid programs.

Money for student aid has been raised through the tax-exempt bond market by at least 12 state-managed loan programs for the last four to five years. Dartmouth, however, is the first private school to go directly to the market and the first to manage a loan program using funds obtained by bond sales.

Several college officials and lobbyists for educational groups expressed concern earlier this week that Congress might attempt to restrict the ability of colleges to raise money for student loans through tax-exempt bonds. But according to congressional sources, it is highly unlikely that colleges will be affected.

A series of new restrictions on the sale of tax-free bonds issued by nongovernmental entities was proposed by the Reagan administration in February and is now under consideration by the Senate Finance Committee.

Although the legislation was directed at the growing use of Industrial Revenue Bonds by private business, it might also have affected bond offerings made by private colleges.

The proposal included tighter notification and registration requirements and a provision mandating financial contributions from the governmental agency authorizing the sale.

Congressional sources said that an administration official from the Treasury Department had proposed that students receiving loans from tax-exempt money be subject to the same eligibility requirements that apply to existing federal loan programs. Reportedly, no such restrictions are contained in language currently before the Senate Finance Committee or in the House version of the same bill.

The Dartmouth move is also significant because the issuing college assumes direct liability for the bond issue. Under existing state programs that use tax-exempt bonds to raise money for loans, liability for the bond issue was assumed by the state government.

Dartmouth, and any other schools that use the bond market, will now be directly liable for repaying the bonds.

With the money raised from its offering, Dartmouth says it plans to start offering loans to students this fall at 11.9 percent interest. Dartmouth will be paying an effective interest rate on its bonds of 10.6 percent. The difference will go to cover the expenses of running the program.

Dartmouth's bonds were rated AA and sold out on the first day of issue, according to college officials. Goldman, Sachs & Co. was the principal underwriter.