The young duo both impressed during the third round at Lumine Golf Club, with Horsfield carding a 66 on the Hills Course while Syme excelled on the Lakes Course by tying the lowest round of the tournament so far with a fine 63.
Horsfield, who joined the paid ranks in May after two years at the University of Florida, followed up scores of 69 and 68 by bagging seven birdies and one bogey during the third day in Spain.
The 21 year old, who qualified for the U.S. Open in 2015 and 2016, sits alongside Ben Evans in second place at the competition’s halfway stage, one shot behind leader Laurie Canter.
With 54 holes left to play the young Englishman knows he still has a lot of work to do if he is to make the European Tour after his first trip to Qualifying School.
“It’s going well so far,” he said. “It’s a marathon, a long old event – I’ve never played a six round event before – so to think that we don’t have one round to go, we have three rounds to go and are only halfway done, is pretty crazy.
“It’s going to be a good few days I hope. I’m just trying to play each shot at a time, you’ve just got to go out there and play your best and see what happens.
“I only turned professional five months ago. I went to the University of Florida, I did two years there, then I turned pro earlier in the summer – I was born in Manchester but moved to Orlando when I was five.
“It depends on how this week ends up but the plan is to try to come back to Europe now. I’m in a good position with three rounds to go but it’s going to be a long week and a lot of golf ahead.
“I played two U.S. Opens as an amateur, played some PGA Tour events and European Tour tournaments and see how good those guys are. They’ve been great learning experiences that I’ve been lucky enough to have.
“I think I’m ready to match that level. The aim is to get a European Tour card this week and keep improving all the time but I’m going to try not to get ahead of myself and just keep doing what I’m doing.”
Also impressing in Tarragona was Scotland’s Syme. The 22 year old, who turned professional in September after representing Great Britain & Ireland in the Walker Cup, was delighted to have signed for his best score as a professional.
“I felt that a good round was coming,” he said. “I played very well over the first two days but I didn’t get much out of them at all, so it was good to see a few putts dropping today.
“It felt like a Moving Day even though we are only halfway through – so it was a bit of a different experience. But I just kept patient and it was a very solid round of golf.
“It was a lot calmer this morning than it has been during the last two days and it was nice to take advantage of them.
“I didn’t feel like I hit it any closer. It’s crazy really to be ten shots different from yesterday. Although I made a silly mistake on 18 yesterday I just played the same – just today the putts were going in.
“I have shot eight under before in a few events, including one pro event, but this is my lowest round as a pro.”
“It is a strange format with there being six rounds and it is very much a marathon. Yesterday I was looking at myself in 95th on the leaderboard but knew that there was still so much golf to be played.
“You have to just keep plodding a few and keep doing my same numbers and same routines and hopefully this form carries on.”
Canter remained calmest on the third day in Spain, with the 28 year old picking up five birdies on his way to a bogey-free 67 that took him one clear of the field, with overnight co-leaders Tom Murray and Andrea Pavan faltering slightly, though Pavan rallied with two late birdies to remain within two shots of the lead.
Moving into the top 20 were European Tour winner Kristoffer Broberg on his return from injury and Irish hotshot Gavin Moynihan, who both shot rounds of 67 to climb to seven under par.
Following the conclusion of the fourth round tomorrow there will be a cut, with the top 70 and ties progressing to the final two days.
On Thursday the top 25 and ties will earn European Tour cards.