There is growing momentum towards the DH becoming universal.
Monday, Jim Bowden of CBS Sports HQ reported that there "is a growing belief amongst NL GM’s that the DH will be instituted for NL as early as 2021."
In May of 2019, MLB commissioner Rob Manfred said that he didn't feel like baseball was close to making the DH universal.
"I don't see anything happening on this outside of the context of collective bargaining," Manfred told the collective media. "So don't worry about it until at least 2021."
What happens in 2021? In December of that year, the MLB's current collective bargaining agreement will expire. There's long been speculation that at that time, a universal DH could be worked into a new CBA. The current CBA doesn't expire until after the 2021 season, though, which makes you think that if the DH comes to the National League, it may not be until 2022. Of course, both sides would probably prefer to hammer out a new deal before the last minute, so anything is possible.
There are many important voices in this debate, but it is worth noting that Manfred himself didn't sound keen on the idea of bringing the DH to the National League.
"Personally, I think the debate about the DH and differences in the brand of baseball is a good thing," Manfred said. "I do think that if we move the DH to the National League we would eliminate or make extinct a brand of baseball. I think that would be an unfortunate result. Nobody plays the National League brand of baseball other than the National League. I think that would be an unfortunate outcome."
Opponents of the DH coming to the National League suggest that a large part of the strategy of the game would be lost if managers didn't have to balance when to use a pinch hitter with taking their starting pitcher out of the game.
With attempting to keep that element of National League baseball in mind, Jayson Stark of The Athletic reported in January that in "behind-the-scene brainstorming sessions" the idea of the DH only being allowed to stay in the game as long as the starting pitcher was discussed. There's no indication that such a solution is especially likely, but it would be a pretty interesting compromise.
The DH has been a fixture in American League baseball since 1973. During interleague play and the World Series, the DH is used when the game is played in an American League Park, but not when a National League team is hosting an American League team. There is a borderline unfair advantage when games are played in the American League park under the current rules, as American League teams employ a full-time DH, while National League teams are left to use a pinch hitter to fill the role.