CLEVELAND, Ohio -- The hunt for free-agent Bryan Shaw is gaining speed.

Two teams have made multiyear offers for the durable Shaw, who spent the last five seasons with the Indians. In those five seasons, Shaw made more appearances (378) than any reliever in the big leagues.

The two offers are for at least three years each.

The identity of the two teams has not been confirmed, but the Mets are believed to be one of them. They have shown interest in Shaw since he hit the open market and Mickey Callaway, Shaw's pitching coach with the Indians, is the Mets new manager.

In three of the last four years, Shaw has led or tied the AL in appearances. This past season he tied Kansas City's Peter Moylan with 79 appearances for the league lead. In those 79 appearances, Shaw went 4-6 with a 3.52 ERA and three saves. He struck out 73 and walked 22 in 76 2/3 innings.

The opposition hit .247 against Shaw last season with lefties hitting .234 and righties .254. The Indians were 51-28 when he pitched.

Shaw is ranked second in relief appearances in Indians' history.

In negotiations to date, Shaw has let it be known to interested teams that he is not interested in waiting for reliever A, B or C to sign before he signs. He has made it clear that if the deal is right, he'll sign whether the market for relievers has been set or not.

When the Indians were knocked out of the postseason in the ALDS, it was clear that they would not be able to keep Shaw based on what he was going to get on the open market. Several online sights projected that he'd sign a three-year deal worth $20 million to $25 million.

The Indians are already paying lefty Andrew Miller $9 million for 2018, while mlbtraderumors.com projects that closer Cody Allen could top $11 million in arbitration in January. To add Shaw's new contract to the back end of the bullpen was probably too much for the Tribe to bear.