The bidding contest for Security National Bank escalated yesterday as Ns&t bYank announced that it intends to offer $62 a share for all of the bank's outstanding stock -- an amount smaller than the price paid for Security's stock in recent transactions.

The offer, which had been anticipated, put a third contender into the fight for control of the prosperous bank -- the city's eighth-largest. On July 22, Washington Bank and Security announced an agreement to merge. That agreement followed Security's rejection of a bid from James Madison Ltd., parent firm of Madison National Bank, to pay $60 a share for up to 44 percent of Security's shares.

On Monday shareholders who own approximately 30 percent of Security's stock banded together in a deal that gives other members of the group first rights of refusal before any of th group may sell shares to an outsider -- an apparent attempt to guard against unfriendly takeover.

Security National Bank late yesterday issued a press release asking its shareholders to refrain from offering their shares until the board of directors has had a chance to make a recommendation on the NS&T offer. The offer needs to be evaluated in light of the agreed-to merger as well as from a legal and regulatory standpoint, the bank said.

NS&T Chairman and President Joseph H. Riley yesterday announced the bank's proposed tender offer, saying that he object is to merge Security into NS&T. Because there is no timetable set for the merger of Washington Bank and Security, it was not clear whether that could occur before an NS&T takeover if the tender offer succeeds.

NS&T reserved the right to not purchase any shares if fewer than 40 percent of the outstanding stock is tendered. On the other hand, it also reserved the right to buy less than 40 percent. The tender offer expires Sept. 1 at 5 p.m.

The offer is subject to normal regulatory approvals as well as the successful completion of a pending holding company reorganizatin of NS&T, which would result in the creation of NS&T Bankshares Inc. The holding company would make the tender offer for Security. Riley said that the combination of the banks would benefit shareholders of each institution.

Security officials said yesterday that the bank had not received a copy of the offer and was not in a position to evaluate the offer.

The tender fight is part of what is expected to be a growing wave of consolidation among area financial institutions. Security has total assets of $136.582 million as of July 1. Its earnings increased 34 percent in the first half of this year after a record year in 1980.