The chairman of the Utah Board of Education said yesterday there was "mutual agreement" between his board and Charles M. Bernardo, who was ousted in 1979 as Montgomery County school superintendent, that it would be best for Bernardo to withdraw from his recent appointment as Utah's state school superintendent.

Bernardo's decision came amid indications that the Utah board that named him two weeks ago had changed its mind. Bernardo disclosed his decision Thursday night, saying he was worried that he would be in a "tenuous circumstance" if he took the Utah post. Earlier Thursday, school board chairman Jay A. Monson said, "There is a possibility now a majority does not exist [in support of Bernardo]."

Monson said in a telephone interview yesterday that Bernardo's decision was "best for education in this state. If a superintendent doesn't have clear support on the board . . . it would be wise for him to do something else."

Earlier, when explaining the decision to pick Bernardo, Monson said that although the Utah board is "a very conservative [board] in a very conservative state" his panel was "very impressed" with Bernardo and "willing to take the risk" of hiring him.

Bernardo's wife indicated that her husband did not want to be interviewed by a Washington Post reporter. Instead, she read a statement in which he said he now preferred to "take advantage of an alternative opportunity combining business and education domestically and abroad." A similar statement was read to a reporter by a secretary at the Falls Church office of Palm Coast, Fla., Inc., a Florida real estate development firm, for whom Bernardo works as a salesman.

During his 3 1/2 years as Montgomery school superintendent Bernardo, who carried a reputation as a liberal reformer, came under heavy fire from the teachers union and parents groups. He was the main issue in the 1978 election in which a conservative slate, pledged to firing him and restoring "basic education," swept to victory.

On May 27 the Utah board reconfirmed its decision to name Bernardo after receiving phone calls critical of Bernardo from people in the Washington area. Monson said yesterday he told Bernardo after that meeting the board had split 6 to 5 in its private vote on his appointment to the $63,000 a year post, even though its public vote was unanimous. Monson said he was one of those originally opposed to naming Bernardo.

Harriet Bernstein, a former member of the Montgomery board, said yesterday that Bernardo told her he had put a $2,000 deposit on a house in Salt Lake City, had given notice that he would leave the house he rents in Rockville, and that his wife had resigned her job as a supervisor in the Montgomery school system.

Monson said Bernardo had asked the Utah board for a "financial settlement" and said the board would discuss the request at a meeting Monday.