Officials from the Tigers, Indians and Rangers have observed throwing sessions by free agent James Shields in recent weeks, sources say, as the 37-year-old remains focused on continuing his Major League career.

Shields was not close to signing with a team as of Monday evening, but his market is expected to take shape over the coming weeks, given the attrition faced by rotations around the Majors.

Shields is probably the second-most-coveted starting pitcher still available in free agency, after former Cy Young Award winner Dallas Keuchel. Unless a suitor emerges soon, Keuchel is increasingly likely to wait until after next month’s MLB Draft before signing.

If Keuchel remains unsigned after the Draft’s conclusion on June 5, his market no longer will be influenced by Draft-pick compensation. Keuchel, 31, then would have the option of signing a one-year deal and re-entering the market next winter, when he would not be eligible to receive a qualifying offer.

The Tigers, Indians and Rangers are not viewed as strong candidates to sign Keuchel, apparently because of his asking price. The Yankees, Brewers and Angels are among the possible landing spots, depending on the health status of their injured pitchers -- and their place in the standings -- when the Draft is over.

Keuchel, who has a 3.31 ERA in 10 postseason games, also could appeal to the Padres, whose rotation consists entirely of starters 27 and younger. Padres officials also are seeking to manage the workloads of Chris Paddack and Matt Strahm, coming off injury-shortened seasons; the addition of Keuchel, a former World Series champion, would do far more than address that.

Meanwhile, Shields’ reliability is a large part of his appeal. Since his Major League debut in 2006, he has posted 10 seasons of at least 200 innings pitched -- including with the White Sox last year. Justin Verlander (11) is the only pitcher with more over that span.

Shields went 7-16 with a 4.53 ERA in 34 games (33 starts) last year. He is 3-6 with a 5.46 ERA in 11 career postseason starts.