After a hot start in 2013, he fell apart, hitting only .187/.287/.282 from May to when the Indians released him in mid-August. After the Yankees signed him no one expected much, but he managed to hit .236/.300/.455 with six home runs in 36 games.
One reason for his struggles was likely Progressive Field's disastrous effect on right-handed hitters. According to Statcorner, the ballpark's Park Factors were 88 for right-handed home runs and 118 for left-handed home runs in 2013. Yankee Stadium is notoriously lefty-friendly, but in 2013, righties had a 122 mark. He might not be able to take advantage of the short porch in right field, especially since he is a dead pull hitter, but the run environment there would still be much better than it was at Progressive.
With everything the Yankees are facing in the offseason and a very limited amount of free agent options, Reynolds has to go to the top of their list for free agent signings. Obviously, everything depends on what happens with Alex Rodriguez and his appeal. Will he be suspended for the entire season? Part of the season? Not at all? Reynolds' eagerness likely stems from his awareness of this situation and is hoping to nail down a contract and a job from a panicking Yankees front office before anything is decided with A-Rod.
Once they have a better idea of how much money they have and how many games they need to fill they will take the appropriate actions. What the Yankees don't need to do is panic and give a large amount of money to a bench player. He has made $5 million, $7 million, and $6 million over the last three seasons, so even if they bump his pay up to around $8-9 million, he would still make less than Kevin Youkilis's $12 million in 2013 without any of the injury history.
Regardless of what happens with the appeal, the Yankees will still need a second third baseman to at least come off the bech. The market really comes down to Reynolds and Michael Young. Everyone else is either much older or Eduardo Nunez.