In a handful of places the NT interprets Jesus’ resurrection (+ ascension) as his exaltation to God’s right hand, which is to say, as his fulfillment of David’s prophecy about the coming messiah in Ps 110:1: “Sit at my right hand until I make your enemies a footstool for your feet.” See, for example, Heb 1:13; 8:1 and 10:12–13. Despite all this, some still insist that it isn’t appropriate to say that Jesus is presently reigning as the davidic messiah, the davidic king (see, e.g., here). One problem with this reading is that it seems to contradict what Paul says about Jesus in 1 Cor 15:25. There Paul says that Jesus “must reign until he has put all his enemies under his feet.” The bit about the “enemies” recalls the end of Ps 110:1 and therefore suggests that Paul thinks that what Jesus is doing now at God’s right hand—the first part of Ps 110:1—can be summarized with the word “reign.” In short, if the NT says that Jesus’ exaltation fulfills the promise of Ps 110:1, then, according to Paul, we must say that Jesus, the davidic messiah, is presently reigning.
That’s the short version of the argument. Here, however, let me add one more note. Some may still wonder whether Paul’s language about “reign[ing] until he has put all his enemies under his feet” is indeed a reference to Ps 110:1, since Paul doesn’t indicate he’s citing Scripture and, as well, since Paul uses slightly different language than what we find in Ps 110:1. In this case, what Paul says here wouldn’t require us to say Jesus is reigning now, since the author could simply be describing something in the future, something different than what David prophesied in Ps 110:1 and, therefore, different from what the NT authors say was fulfilled at Jesus’ exaltation.
That sort of reading is possible, but unlikely. Just because Paul doesn’t use an introductory formula (e.g., “as it’s been written”) doesn’t mean he isn’t intentionally recalling Ps 110. He sometimes simply cites an OT text (e.g., Gal 3:6), often verbatim but not always. We call these latter instances allusions. What suggests that Paul alludes to Ps 110:1 here is that he uses language that is identical to Ps 110:1 and, moreover, language that is found only in Ps 110:1. That is, not only does he use language found in Ps 110:1 (“place,” “enemies,” “feet”) but this language is found together nowhere else in the entire Greek OT. What’s more, the parts of Paul’s presumed allusion that use different language don’t point away from the presence of Ps 110 but rather to the influence of Paul’s present context. That is, Paul’s changed the direct discourse of Ps 110:1 (“I will make your enemies a footstool for your feet”) into indirect discourse (“he will place his enemies under his feet”) to seamlessly incorporate the text into v. 25, and, as well, he’s substituted “footstool for…feet” for “under…feet” and added “all” before “enemies” probably because of the influence of Ps 8, which Paul recalls in 1 Cor 15:27. In fact, it is his certain allusion to Ps 8:6 in v. 27 that suggests, even more strongly, that Paul’s language in v. 25 intends to recall Ps 110, since everywhere that Ps 8:6 is discussed in the NT it is combined with Ps 110:1 (see, esp., Eph 1:20, 22 and Heb 1:13; 2:7).