If Steven Gerrard’s post-Liverpool future lies in Major League Soccer, there will be no shortage of clubs – with LA Galaxy leading the line – interested in securing the 34-year-old midfielder for the final years of his career. Liverpool have sold out stadiums in the United States on friendly visits and Premier League television coverage is attracting growing audiences. The acquisition of Gerrard would be a major coup for the MLS as it heads into its 20th season.

The MLS is growing nicely — attendances and interest are on the rise and the standard of play is gradually improving – but the league has lost some of its stellar appeal in the past few months and Gerrard would fit the bill as a name capable of generating excitement. While MLS has a salary cap, the Beckham Rule allows for three designated players whose salaries do not count fully against the cap and who can be paid lucrative deals.

Finding the right players who can still perform at a good level has not been easy. Thierry Henry, the biggest name in the league since David Beckham left Galaxy in 2012, ended his four and a half years with New York Red Bulls in November and Landon Donovan, the league’s best known American player, has also retired.

Toronto hoped Jermain Defoe would boost their profile and performance but he is now looking to return to England after a disappointing spell with the Canadian club. Orlando City have signed the Brazilian Kaká, formerly of Milan and Real Madrid, and New York City, who join the league this year, have signed the former Spain and Barcelona striker David Villa but the Frank Lampard signing seems to have backfired.

The protracted Lampard affair has left a bitter taste for many American fans and means there may be wariness about bringing in another English player near the end of his career but Liverpool’s huge fan base in North America makes Gerrard an attractive proposition.

Gerrard may not have the mainstream media appeal of Beckham but his widely respected attitude to the game and leadership skills would appeal to MLS clubs who also expect their designated players to act as mentors to American youngsters and be ambassadors for the club.

Beckham’s former club Galaxy have been the most strongly linked with Gerrard. The MLS champions are in the market for a big-name signing given the Irish striker Robbie Keane is their last remaining well-known foreign player and he is in the final year of his contract.

The Galaxy coach and general manager Bruce Arena, who had dismissed rumours of a Gerrard move to Los Angeles, responded with a “no comment” to the LA Times when asked about possible negotiations while the club president, Chris Klein, added: “We don’t have anything yet. We don’t comment on players that are not with our club. We’ve been pretty consistent with that all along.”

The club, owned by the entertainment giants AEG, showed with their signing of Beckham they have the cash to spend on top players. While Beckham made Galaxy the glamour team of MLS, they will face a new challenge in 2017 when a heavily financed second LA team joins the league and the recruitment of Gerrard would help them tackle the risk of being overshadowed by big-spending new neighbours.

If Toronto can free up Defoe’s $6.18m (£4.01m) a year wages they might be able to tempt Gerrard. The Seattle Sounders are MLS’s best supported club with an average attendance of 43,734 last season and have the former Fulham and Tottenham forward Clint Dempsey and the former Newcastle Nigeria international Obafemi Martins in their lineup but their artificial surface could be a deterrent.

If Gerrard does make the move to MLS he is unlikely to be the only European player heading in that direction. Keane said recently he regularly receives calls from players asking about opportunities in the States. “There a lot of players that I know personally, big players, who want to come now. I can’t tell you who they are but it tells me how far this league has come,” he said.

An MLS spokesman declined to comment on the reports linking Gerrard with a move to one of its clubs but said: “Steven Gerrard is a world-class player and soccer fans throughout the world have admired his talents for many years.”