After failing to land Yu Darvish with a five-year contract offer, the Twins are looking at several other free-agent pitchers, as well as exploring trades with multiple teams, according to several sources with know­ledge of the club’s search. One pitcher the Twins have targeted is Rays righthander Chris Archer, and the club made a trade offer as recently as two weeks ago, a source confirmed. Tampa Bay has some interest in outfielder Max Kepler as part of any package for Archer. Kepler, with a year and 153 days of service time, is under team control for five more years, which appeals to the Rays. He turned 25 Saturday and batted .243 last season with 19 homers and 69 RBI. Archer, 29, is 51-63 during his six-year career. He has struck out 734 batters over the past three seasons, but he has not won more than 12 games in any year and went 10-12 last season with a 4.07 ERA. Still, Archer can pitch like an ace when he is on. He had Twins fans buzzing during a May 26 outing at Target Field when he struck out 11 over 7 ⅔ innings in a 5-2 Rays victory. If the Rays are willing to move Archer, the Twins likely will have to top other offers from across the majors. He will make $13.75 million over the next two years with club options for $9 million in 2020 and $11 million in ’21. His affordability makes him desirable, and puts Tampa Bay in position to demand a lot in return. Jake Arrieta, Lance Lynn and Alex Cobb are considered the best of the remaining free-agent starting pitchers. The Twins have expressed interest in Lynn, 11-8 with a 3.43 ERA with St. Louis last season, as well as Cobb, who was 12-10 with a 3.66 ERA with Tampa Bay. The Twins’ chances of signing Arrieta, the 2015 NL Cy Young Award winner with the Chicago Cubs, are remote. The Twins could also look at lefthander Jaime Garcia, who joined them for a week last July before the club pivoted and dealt him to the Yankees. Another name to watch is Chris Tillman. The righthander is a two-time 16-game winner with Baltimore but went 1-7 with a 7.84 ERA last season, when he battled a shoulder issue. But Tillman is 29, and the Twins might view him as a bounce-back candidate. Upgrading the starting rotation was the Twins’ top priority this offseason and — outside of signing Michael Pineda, who had elbow ligament replacement surgery last July — they have been unable to address it. They went on the record about their interest in Darvish from the outset of free agency, which was unprecedented for the club. They were ready to invest in top tier pitching, as long as it made sense to them.