Andy Murray, simultaneously relieved and disappointed after enduring two long matches of a satisfactory grand slam comeback at the US Open, says he is desperate to return to Davis Cup action in Glasgow in a fortnight.

While the Scot is equivocal about the controversial reforms that threaten to marginalise the oldest team event in sport, a chance to play again in the city where he was born – for Great Britain in the tie against Uzbekistan, which starts on 14 September – seems irresistible.

“Potentially it’s the last time I’d get to play competitive tennis in Scotland, so I’d like to do that,” he told the BBC. “I’d have to chat to my team because this is obviously a very important period in my rehab and my long-term strategy.”

Murray, who last played Davis Cup when Argentina won in Glasgow in 2016, left New York spent but in one piece after Fernando Verdasco beat him 7-5, 2-6, 6-4, 6-4 in three hours and 23 minutes in sweltering heat on Wednesday afternoon. That followed Murray’s impressive four-set win in three hours and 17 minutes against the competitive Australian James Duckworth in similarly tough conditions on the opening day.

Murray and Kyle Edmund missed Great Britain’s fighting loss to Spain in February, which was notable for the impressive debut of Cameron Norrie. Edmund, Britain’s No 1, is yet to declare himself available after a non-specific illness cut him down here in the first round, so Murray and Norrie – who went out in the second round on Wednesday - are favourites to lead the line.

Defeat at Flushing Meadows did not leave Murray distraught, as it used to in the distant past. He is comfortable in his life away from the court, a husband and father of two daughters. It is understandable that he and Kim have decided their growing family should form the increasingly important focus of their lives, as his playing career enters its autumnal days.

Yet the ending to the fourth set – especially the last game, which Murray extended to 12 minutes – suggests the hunger to achieve more is still strong.

Whether or not he adds to his three slam titles is the great unknown. Bookmakers put the odds at 8-1. He has returned at another moment of flux in the game, with this tournament again examining the credentials of his long-time rivals as well as the young contenders. On the evidence so far, Murray’s task is more challenging than when he was at his peak, whatever the state of his fitness.

Where he can determine the direction of his game is to cut down the time on court by taking more risks, as he seeks to inject more attack into his grinding defensive game. His occasional forward sprinting is impressive but he is still struggling with his lateral movement.

And he has yet to rediscover the swinging of the torso behind his forehand, particularly, that made it such a lethal weapon from mid-court before his flexibility was compromised. Instead, he has to rely on the power of his arms alone mostly when going for winners, and that turns him square to the net, leaking power from the shot.

The core of his game – and currently a weakness – is the serve, which let him down against Verdasco, who only had to wait for the first one to pass harmlessly outside the box to pounce on the second.

Murray’s own analysis of his performance immediately after the match was as honest and perceptive as any. “Some of the tennis I played was some of the best I’ve played since I had the surgery or since I came back,” he said of a four-set match – his second in three days in his first slam for nearly 14 months – that included humid, 36C heat.

“To still be doing as well as I was at the end of the match, considering the lack of practice and matches that I’ve had, was positive. At the end, when my back was against the wall, I came up with some good tennis to make it close and interesting and almost got myself back into it.

“I did all right. I chased balls down right to the end of the match. I wasn’t giving up on points. It wasn’t the most comfortable I felt on a tennis court. I got through it and fought right to the end. But there were also periods in the match, especially in the first set, where I really didn’t play particularly well. There were too many ups and downs for my liking.”

While Murray agrees it is “completely normal” for people to speculate about his future, it looks likely he is not going to leave either soon or quietly.