The New York Red Bulls are on their way to one of their best seasons in franchise history. They claimed the top spot in Major League Soccer’s Eastern Conference and have a strong chance at winning their first ever title.

Fans have been eager to support the team in all the traditional ways, like waving scarves and screaming their heads off. But a surprising number have contacted the team because they think can help in a more practical way: by suiting up for the team.

Never mind that the Red Bulls have the MLS’s leading goal scorer, Bradley Wright-Phillips, and a handful of other players who represent national teams across the globe. These fans tune into the games, cheer on the team and leave convinced that they could hold down on a spot on of Major League Soccer’s leading teams.

“Folks have to remember that [the players] are elite athletes,” says Marc de Grandpre, the Red Bulls’ general manager. “It takes a lot of hard work and dedication to get to that level.”

Some fans seem to think otherwise. This season, the Red Bull’s social media accounts have been inundated with requests from spectators—and not always ones with strong soccer pedigrees—to try out for the team. Fans send tweets and messages on Facebook. Requests even come in on Instagram and Snapchat.

The Red Bulls’ Bradley Wright-Phillips is the leading scorer in the MLS, but some fans still think the team could use their help. Photo: Corbis via Getty Images

Sebastian Milo is one of the fans who reached out to offer his services to the Red Bulls, who lost 1-0 Sunday in the first leg of their playoff series against the Montreal Impact. Milo, 18 years old, isn’t on his own college team, but played soccer when he was younger. He didn’t receive a response from the team.

“If the teams allow them to [try out],” Milo said, “people could make something of themselves.”

The Red Bulls aren’t the only team to receive these unlikely tryout requests. Though MLS is the top U.S. professional soccer league, the level of talent is generally considered to be a step or two below the top European leagues. That seems to have convinced a growing number of fans that they deserve a shot at cracking the roster.

Jonathan Chan, a 20-year-old fan, played soccer in high school, though he admits he he “wasn’t exactly what you’d call the best” player on the team. But when he flipped on an NYCFC in May and saw the team losing 7-0, he thought the team that boasts international superstars like David Villa and Andrea Pirlo was missing something.

“I was being serious,” said Chan, who sent a tweet to the team asking for an audition. “Having skill and being good is important, but you have to have heart, you know?”

MLS clubs aren’t opposed to offering opportunities to talented locals. The Red Bulls youth academy and training programs reach up to 40,000 kids in the Tri-State area, Grandpre says. The best of them might have a chance to work their way up through the Red Bulls’ team in the United Soccer League, which operates as a de facto minor-league for MLS clubs.

Some teams have even held open tryouts for these feeder clubs, thought executives admit it’s a longshot proposition for any attendee. The Seattle Sounders and LA Galaxy, for example, will both hold tryouts for their USL teams in the coming months.

Those tryouts yield strong interest. Possibly too strong. Last year’s tryouts for the Galaxy were an accomplishment: For the first time, nobody who made it to the trial with the team got hurt. When they held open tryouts for the first time a few years ago, a middle-aged man in his 40s from Arkansas trekked across the country and announced he would be auditioning at goalkeeper.

“He actually played really well,” said Zach Wells, a former MLS goalkeeper who is the Galaxy’s Manager of Brand Development and Fan Management. Since then, the team has placed age restrictions on the tryouts. “Obviously we recognized that wasn’t getting to the point where we wanted to be.”

That so many fans seem to think they can play MLS-caliber soccer shows they still may be a bit skeptical about the product on the field. Although the league has grown on its knack for attracting some international stars, albeit usually aging ones, nobody would confuse the clubs with FC Barcelona or Manchester United.

Still, what’s encouraging from the league’s perspective is seeing so many people interested. The average attendance for the league’s 20 teams this season reached a record high of 21,692 per game, a 40% increase since 2006.

That growth is reflective of the quality of the play in the league, says Victor Suchy, a Red Bulls season-ticket holder since the franchise’s inception, back when the team was called the MetroStars. When he first bought season tickets, he thought he might be good enough to get on the pitch.

Now he’s more skeptical. “Not just anybody can make the cut,” Suchy said.

Write to Andrew Beaton at andrew.beaton@wsj.com