One of those teams, the Twins, has some positive recent developments to hang their hopes on, while the other, the Red Sox, are still scuffling and come into the series the losers of five straight and in last place in the American League East standings.
The Twins started the season 0-4, but began to put the pieces back together and are 5-11 since then after losing to Tampa Bay, 6-2, Sunday.
They got good news when right-hander Nick Blackburn, who has been idle since April 14 with a sore shoulder, made it through his bullpen session Sunday with no problems. He is slotted to start on Tuesday in the second game of the series.
Also, Joe Mauer, who suffered through an injury-dominated 2011 season, has started all 16 games this season, certainly welcome news since he is the unquestioned cornerstone of the Twins.
Center fielder Denard Span is one Twin who sees reason for optimism.
"I feel like we've had a good road trip," Span said. "We easily could have won three out of four in New York. I like what I've seen from this team after starting 0-4."
Manager Ron Gardenhire still is keeping things in perspective.
"Getting back to the old way we play?" Gardenhire said when asked about the Twins' improvement. "I don't know. The old way we play is we win more than we lose, and we've got to get back to that part of it."
The Red Sox are probably happy to leave the miserable weather in the Boston area that forced the postponement of their Sunday night game against the Yankees and a shuffling of their scheduled pitchers.
Losers of 10 of their first 14 games, they will seek to start some kind of turnaround in Minneapolis, with veteran Jon Lester (0-2, 5.82 ERA) getting the start against the Twins' Jason Marquis (1-0, 7.20 ERA).
Red Sox: Beckett trying to get on track
The Red Sox are expected to follow Lester with fellow veteran Josh Beckett in Game 2 of the series. Beckett is 1-2 with a 5:03 ERA.
Twins: Mauer moving up hit list
Mauer has 1114 hits and needs 50 to tie Cesar Tovar for ninth on the Twins all-time list.
Marlon Byrd, acquired from the Cubs Saturday to shore up the Red Sox's outfield depth, was scheduled to start Sunday's game against Yankees left-hander CC Sabathia, but it is uncertain whether he will start Monday against the right-handed Marquis.