UK universities are bracing themselves for a last-minute battle for new recruits in what is expected to be the most competitive A-level student recruitment round to date.

Hundreds of thousands of 18-year-olds will pick up their A-level results next Thursday. For those intending to go to university it will be a buyers’ market, with many of the country’s leading institutions vying to attract students to take up places on tens of thousands of courses that are still unfilled.

Competition for students is intense after the lifting of the cap on student numbers and because of a dip in the number of 18-year-olds in the population. There are additional concerns this year that the number of EU students will fall because of Brexit uncertainty.

According to the university admissions service, Ucas, places were still available on more than 30,000 courses in 350 providers across the UK this week. More are likely to become available through the clearing process once results are released and universities have a clearer idea of numbers.

Many of the most popular and sought-after institutions in the UK have already posted vacancies. On the Ucas clearing site on Thursday, Bristol University listed more than 130 courses with places still available, Exeter had 270 and Leeds 260.

Among the subjects still up for grabs were law, medicine and English. After years of solid growth, there has also been a significant drop in the number of applications to study maths this year, which means places are still available.

Parents in England are being urged to take time off work next week for results day, when a record number of students are expected to search for a university place through clearing.

Richard O’Kelly, the head of data and analysis at Ucas, said: “Clearing acceptances have been on the rise for several years. There’s been a 50% increase in the last decade – from 43,890 in 2008, to 66,770 in 2018. We don’t think it’s unreasonable to see more than 70,000 students get their place through clearing this year, 80,000 even.

“University admissions is a competitive market, especially with the continual decline over the past few years of the UK’s 18-year-old population, and this is undoubtedly true for clearing too. Displays at train stations, pre-trailer cinema adverts, plus constant boomerangs, gifs and stories across social media right now are testament to that.”

While clearing is an effective way of matching students who may have just missed their required grades to vacancies elsewhere, it can result in students having to rethink their plans, studying at an alternative university in a different location and possibly a different subject.

Julie Kelly, the head of the student centre at Hertfordshire University, said: “If your son or daughter ends up in clearing they are going to need your support. I would recommend booking the day off work, either to be on hand for support or to celebrate their results.

“Luckily most universities have some form of virtual open day, which helps students who can’t visit in person; these are a great resource to review a couple of days before the results come. Ask your child to ensure they have a list of the universities they are looking to approach, the courses they want to study and the universities’ clearing hotline numbers.”

Despite a 1.9% fall in the overall 18-year-old population in the UK, Ucas said a record number of young people applied for university this year, up 1% on 2018. Applications from EU students also held up, but admissions teams fear many of those applications will not result in students taking up places to study from September because of the uncertainty over Brexit and continuing access to student loans.

Mike Nicholson, the director of undergraduate admissions and outreach at the University of Bath, warned: “This year and next year are going to be very tough for higher education. Next year we will know whether we are in or out of Europe. If we are out, that will take a whole lot of students out of the system.”

Universities who have undergone massive expansion in recent years, with expensive building programmes, will be hit hardest by contracting student numbers. Students who might once have gone to a mid- or lower-tariff institution will be able to trade up with places still available at many Russell Group universities. As a result, some universities face severe financial difficulties.

After the uncertainty of previous years with the rollout of new qualifications intended to add rigour to the exams system and bring it up to date, the A-level reforms are this year largely complete and results are expected to be stable.

Philip Nye, the external affairs manager at FFT Education Datalab, said one of the most significant changes appeared to be a drop in the number of entries for A-level maths, from about 91,000 in 2018 to 86,000 this year. Further maths was also down.

“This is the first cohort who will have taken the new GCSE in 2017. The Mathematics Association was warning if students have struggled with the increased scope at GCSE that might put them off A-level.”

Entries in English language and English literature have also declined, while modern foreign languages, which have plummeted in popularity for years, appear to have just about held up this year, said Nye, though early Ofqual data showed that fewer than 3,000 students were entered for German A-level.