The decision was made after a meeting of MPs from all parties on Wednesday afternoon.
Senior public servant Sir Maarten Wevers will carry out the review, and is expected to make a decision within two weeks.
Speaker David Carter said Mr Matthews rang him and offered to stand down last night.
The move comes after Opposition MPs questioned Mr Matthews' handling of a fraud case in his former role as Secretary of Transport.
Mr Matthews led the transport ministry while former senior manager Joanne Harrison stole $750,000 through fake invoices to companies which did not produce any work.
Harrison was jailed for three years and four months in February this year.