Unfortunately for the Wimbledon champion, he was actually standing across the net in their quarter-final showdown and for over 90 minutes on Tuesday he kept being dragged into a succession of fast-and-furious rallies that left him a set and a breakdown.
Luckily for Alcaraz, he eventually produced the goods on the crunch moments to end the spellbinding contest with a 5-7, 6-4, 6-2, 6-2 victory.
As Alcaraz desperately chased after the "laser beam" winners flying off Paul's racket in the first set, the Spaniard often found himself sliding and slipping over around the threadbare baseline with the match taking place under a closed roof on another sodden day in London.
However, any hopes Paul might have harboured of ending his country's 21-year-wait for a men's Grand Slam champion at the All England Club this weekend were eventually snuffed out.
"In the first set and at the beginning of the second set, (it felt like) I was playing on clay, with big rallies and every point going over 10-15 shots," Alcaraz told the crowd after winning the battle of the two most recent Queen's Club champions.
"I had to stay strong mentally when I lost the first set, it was difficult for me. It was a long journey for me, it was a long match so I was happy to find the solutions (to win)."
With Spain taking on France in the Euro 2024 semi-finals later on Tuesday, diehard football fan Alcaraz was probably hoping that he would be back in Wimbledon's plush locker room in double quick time, having extended his perfect Grand Slam record against Americans to 7-0.
It was not long before Paul provided him with a harsh reality check.
Alcaraz earned four break points in the opening game, including a pulsating 14-shot rally which he capped by smashing the ball away with his back almost turned to the net. Yet all four of those opportunities went begging.
The third seed was kept on his toes for 17 captivating minutes as he was forced to save four break points while being stretched to nine deuces before he finally managed to hold serve on his sixth game point - and all this effort just to get to 3-3 in the first set.
It was no wonder he was shouting "Vamos" and raising a left-arm salute as the crowd jumped to their feet to give both players a standing ovation.
But Paul kept producing the delicious drop shots, ferocious forehands and luscious lobs to make life uncomfortable for the Spaniard.
As he drilled a backhand down-the-line winner past Alcaraz's outstretched racket to seal the first set and then broke for a 2-0 lead, he kept drawing roars of approval from the Court One crowd.
But just when he might have started dreaming that he might finally become the first American man to reach the semi-final at Wimbledon since John Isner in 2018, Alcaraz said enough was enough and quickly got to work to douse the fireworks.
Paul managed to save 19 break points. But unfortunately for him, he dropped serve eight times which was enough for an all-court tactician like Alcaraz, who has won majors on three different surfaces, to seal victory.
"He can play some seriously amazing, amazing tennis. Half of the job when you're out there is not to let him win one of those crazy points because when he does, he kind of gets on a roll," Paul said.
After being kept on court for just over three hours, the Spanish third seed won his 12th match in a row at the grasscourt major when Paul swiped the ball wide.
"I (always) believe I can come back, even if I am struggling or am in trouble. I believe I can find the solutions even if the opponent is playing great tennis. I believe in myself all the time," added Alcaraz, who will next meet 2021 US Open champion Daniil Medvedev in a rematch of last year's semi-final.