The U.S. Department of Labor said yesterday it will begin its own investigation into Montgomery County's federally funded job-training program, following allegations that funds were misused and employes were improperly laid off.

County Attorney Paul A. McGuckian announced the federal probe to County Council members even as they were considering a request by the county's Merit System Protection Board for county funds to conduct a six-month investigation into the same allegations. McGuckian said he requested the federal investigation because it would "provide an absolutely impartial and expert appraisal" of how the county runs its job-training program.

Labor Department spokesman Jack Hord in Philadelphia confirmed the federal probe, which he said would be conducted by the Inspector General's Office of the department. "It will be an investigation into all of the allegations that have been made," said Lord.

The allegations center around the federally funded Comprehensive Employment and Training Act program, which was merged into the county's Office of Economic Development in January 1982. Ioanna Morfessis, director of the office, said federal fund reductions in the CETA program forced several layoffs, but some of those employes have complained that the layoffs could have been averted had the money been handled properly.

The council yesterday approved $30,000 for the merit board to begin its probe into possible improper use of CETA personnel. The board has said it will eventually need $100,000 for the job--including hiring Baltimore investigator Howard S. Rosenstein at $80 an hour.

McGuckian said he told the board in a closed hearing last week that "the going rate for investigators . . . was $25 to $35 per hour."

But board chairman Fernando Bren said that Rosenstein, a former IRS agent, is also a lawyer who is already familiar with county government from his previous investigation of hiring practices in the liquor department. That investigation cost more than $500,000.

The council directed the board to start with $30,000 of the requested $100,000 after council member William E. Hanna Jr. said the allegations were too vague. Other council members said the scope of the merit board's probe could now be reduced because the federal government is also investigating the matter.

McGuckian also told the council that he is retaining an outside lawyer, Roger Titus of Rockville, to represent the county government during the merit board's investigation of CETA. McGuckian said he considers the hiring of an outside attorney necessary because of the deteriorating relationship between himself and Bren about how the investigation should be handled.

The county will pay Titus, the former lawyer for the City of Rockville, $105 an hour and $60 an hour for an associate. Some council members expressed concern about the already escalating costs of the probe.