Norris has made 12 starts this season and pitched poorly only twice. His last three starters -- two earned runs in 19 innings -- offer hope that he has fulfilled the promise the Astros had for him when he made his debut four years ago.
He's also about to start making some serious money and is two years from free agency, so do the Astros make a commitment to Norris? Or do they trade him to a contender for a promising prospect or two? For the last three years, that's the course they've taken with virtually every veteran player in the system.
Lance Berkman and Hunter Pence were traded. So were Carlos Lee and Wandy Rodriguez, plus plenty of others.
As a result, two things have happened. First, the Astros have lost their share of games. Four straight losing seasons is their longest streak since LBJ was in the White House.
Second, their farm system has been transformed from one of baseball's worst to one of its best.
At some point, the unloading of players will stop and the reconstruction of a franchise will end. When is the time to begin the next step? In other words, how close are the Astros to being good again?
Here's a rosy scenario: They're close. Houston will take a five-game winning streak into Monday's game at Anaheim primarily because its starting pitching posted a 2.67 ERA.
It's a reminder that baseball isn't all that complicated. When teams pitch well, they're in every game. At a time of so much parity, it's the small things that separate the top two dozen or so teams.
The Tampa Bay Rays have averaged 92 victories the last five years with some of the lowest payrolls in the game due to the core belief that good pitching and sound fundamentals (defense, baserunning, not giving outs away) can overcome a lot of deficiencies.
If you look at the Astros from a certain angle, you see the light at the end of the tunnel. Besides Norris, 22-year-old right-hander Jordan Lyles, a former first-round pick, appears close to becoming a reliable Major League starter.
Best of all, some of those highly rated Minor Leaguers are having very good seasons. Right-handers Asher Wojciechowski, 24, and Michael Foltynewicz, 21, are at Triple-A and Double-A, respectively, both being fast-tracked to the Major Leagues. Outfielder George Springer, 23, has 17 home runs and 17 stolen bases at Double-A and seems headed for a Major League debut later this season.
If the Astros take a college pitcher -- for instance, Oklahoma's Jonathan Gray or Stanford's Mark Appel -- with the top pick in this year's Draft, they'd add to a growing collection of power arms.
Meanwhile, three of their Major Leaguers -- 26-year-old catcher Jason Castro, 23-year-old second baseman Jose Altuve and 23-year-old third baseman Matt Dominguez -- have played their way into the club's future.
All of a sudden, Houston seems to have the makings of a foundation. Here's a reminder that young players don't come with guarantees. Patience is required. The Kansas City Royals had baseball's best farm system three years ago, and so far, all of that talent hasn't translated into success at the Major League level.
But it's the only way to build a franchise, and the Astros have made enough progress that it's fair to consider a timetable for the next step.
Norris has two more seasons of arbitration eligibility before reaching free agency. If he continues to pitch well, his salary will rise quickly. Luhnow may see enough young pitching in Lyles, Foltynewicz, et al, that he decides to shop Norris for more young talent.
But Luhnow may decide to let fans know that things are about to change. Besides, money may be less a deciding factor than how close he thinks the club is to winning.
Houston has a payroll of under $28 million this season and only $6 million in commitments for 2014. When Luhnow decides the time has come to shape a contending Major League roster, he should have the money to fill a hole or two.
OK, back to that history of clubs having back-to-back No. 1 picks. The Astros will join the Nationals and the Rays as the only other franchises to have done it.
The Rays were in the postseason four months after the second of those No. 1 picks -- infielder Tim Beckham -- was selected in 2008. He hasn't played in the big leagues yet, but Tampa Bay's 2007 No. 1 draftee, left-hander David Price, pitched five times down the stretch in '08 and five more times in the postseason that year, helping the club reach the World Series.
The Nationals took Stephen Strasburg in 2009 and Bryce Harper in '10, and both of them were contributors on the team that had the best record in baseball last season. Plenty of other smart Draft-day decisions went into building the Rays and Nats, and their general managers did a nice job of shaping their rosters.
The Astros may not be at that point yet, but they may not be as far away as people think. That's why the way they handle Norris is about more than one player.