The Professional Air Traffic Controllers Organization, expressing some contrition for its six-week-old nationwide strike, asked the Federal Labor Relations Authority yesterday to send government negotiators back to the bargaining table.

The union also said in its first public statement on the matter that it would accept arbitration of unresolved issues if no agreement is reached in a new round of talks.

The union's plea for a negotiated settlement came during a 2 1/2-hour hearing on a PATCO decertification recommendation issued Aug. 14 by John Fenton, FLRA administrative law judge.

The government attorneys objected to the union's request, saying that new talks would only compromise the Reagan administration's handling of the illegal strike in which nearly 12,000 controllers have been fired.

"If the Authority were to honor such a request, it would be sanctioning strikes by federal employes," said Dolph Sand, an attorney representing the Federal Aviation Administration, the controllers' employing agency.

"Speaking for this administration, we insist that the law be enforced, and it is our unequivocal position that enforcement of the law requires nothing short of permanent revocation of PATCO's exclusive representative status and its permanent debarment as a labor organization . . .," Sand said.

PATCO officials said privately yesterday that they were shocked by the government's response. They said the compromise plan they presented to the FLRA yesterday was intended to salvage some of their union's dignity while giving the administration a way out of its hardline position that the strike is over, the strikers are fired and there's nothing left to talk about.

Transportation Secretary Drew Lewis had said earlier in the dispute that he would resume bargaining if ordered to do so by a court or other quasi-judicial body. And PATCO officials frankly admitted in private conversations yesterday that they had hoped their request for resumption of talks and possible arbitration would break the deadlock.

"Both sides have been stupid in this thing," said a key PATCO official. "They've admitted their stupidity, indirectly, at one time or another. There's no need to continue this when you have 12,000 jobs and the public's safety at stake.".

PATCO lawyers also said on the record yesterday that the union, and the government, erred.

"There's some truth to the criticism that PATCO and the FAA made some mistakes and that the public has been made to suffer for it," the union's attorney, Richard J. Leighton, told the three-member FLRA panel. He said that by ordering a resumption of talks, the authority would be taking advantage of an "opportunity to be creative" in setting matters aright.

Leighton asked that talks resume by Sept. 23 and that an arbitrator or arbitrators be brought in if no agreement is reached within a month..

But the government attorneys argued that Leighton was asking for something the union denied the government and the public when it went on strike Aug. 3 -- time. On July 31, several days after the union's members overwhelmingly rejected an earlier tentative pact with the FAA, "PATCO gave us less than 72 hours" to reach a new agreement, Sand said.

"Let there be no mistake as to what PATCO is seeking here. It is seeking a result that is advantageous only to PATCO," the FAA attorney said.

Though the government argues that the FLRA can only decertify the union "under the circumstances of the case," FLRA officials say the authority has a number of options -- so-called "remedies -- it can use to discipline the union if it finds that further disciplinary action is warranted.

The authority is expected to make a decision within several days, according to union and government officials.