American Telephone & Telegraph Co., mounting the first hostile takeover in its recent history, yesterday went over the heads of NCR Corp.'s balking directors and asked the company's stockholders directly to sell their shares in a $6.1 billion buyout.

The $90-per-share offer, effective today, came shortly after NCR's board voted to reject AT&T's proposal, first made public over the weekend. The board also moved to strengthen its takeover defenses. But NCR directors also said they would consider a sale to AT&T at a share price of $125 or more, which would add about $2.4 billion to AT&T's cost.

AT&T's cash offer to shareholders comes at a time when such hostile takeovers, Wall Street's signature work in the 1980s, are all but nonexistent due primarily to a drying-up of financing and uncertain economic times.

Industry analysts offered conflicting assessments on AT&T's prospects for raising money to finance the buyout. Some said a company of its stature would have no trouble, but others noted that banks in general are trying to cut back on loans.

NCR Chairman Charles Exley presented his board's response to his AT&T counterpart, Robert Allen, during a one-on-one, 45-minute meeting over coffee at a New York law firm yesterday.

NCR later released an angrily worded letter from Exley to Allen, saying that NCR's board would not "sacrifice a successful American computer company in order to bail out AT&T's failed strategy" in computers.

But Allen maintained in a letter to Exley that the real issue was not the fit of the two companies but price. He asked how Exley could offer NCR for sale at $125 but maintain that a sale at $90 would be a "strategic disaster."

Many analysts suggested that NCR will eventually be sold. "I would pretty much say this company is going to go down," said Geoffrey Johnson of Argus Research Corp. New York, saying that cash offers have generally succeeded in the past. Others saw the offer merely as a way to increase pressure on NCR's board.

AT&T's bid is widely viewed as an admission of defeat in its six-year effort to build a viable computer business following its emergence from the Bell System breakup of 1984. Industry analysts estimate that AT&T has lost $2 billion in that effort.

AT&T has said it wants NCR to retain its identity and current management and take over AT&T's computer business. But Exley has said he would resign if AT&T succeeds in gaining control of his company, raising questions of how valuable the company would be.

"If Exley takes all the top people with him, they're going to end up with warehouse space," said John McCarthy, director of professional systems research of Forrester Research Inc. of Cambridge, Mass.

In its core business of long-distance phone service, AT&T has turned starkly more aggressive in the past two years, slashing prices and slowing, if not stopping, a slide in its share of the $50 billion-plus long-distance market. "It's a new AT&T," said Bradford Peery, chairman of Brad Peery Inc., an investment banking firm in Mill Valley, Calif.

The tender offer, a Wall Street transaction in which shareholders are solicited to sell their shares at a specific price, will expire at midnight on Jan. 4, AT&T said.

The deal AT&T offered to the board provided for payment in AT&T shares. The tender offer to the shareholders, however, is for payment in cash. It would require AT&T to arrange some sort of outside financing for part of the $6.1 billion total.

News of AT&T's designs on NCR, released on Sunday, drove NCR stock up almost $25 a share in trading Monday. The price continued to rise yesterday, gaining $3.12 1/2 to close at $86.62 1/2, before AT&T announced it would go to the shareholders.

Exley and his board have various defensive maneuvers still open to them, including finding another buyer for the company -- an NCR lawyer said yesterday that the company had received calls from several foreign and domestic companies about this possibility. Taking NCR private, with a private group of investors buying up its shares, would be another option.

Yesterday, NCR's board also amended its "poison pill" and other antitakeover devices. The moves, which are likely to be challenged by AT&T in court, were designed to block any takeover that does have the prior approval of the NCR board.

Staff writers David A. Vise and Robert J. McCartney contributed to this report.