The 20-year-old is currently on international duty with Norway Under-21s as they prepare for a European Championship qualifier against Latvia in this month's international break.
Since making his debut in September 2020, Arnstad has made 54 appearances for Anderlecht, scoring once during that three-year period as a first-team player at Lotto Park.
The midfielder gained a wealth of European experience last term, playing 11 matches in the Europa Conference League as Brian Riemer's men reached the quarter-finals before losing to AZ Alkmaar of the Netherlands.
Speaking to Norwegian news outlet VG ahead of a crucial clash against Latvia in Under-21 Euro qualifying, Arnstad confirmed that Burnley had made an unsuccessful attempt to sign him in the summer transfer window.
"They did not agree [a deal], but of course it is cool to have some interest," Arnstad told the media during an in-person interview at a player hotel in Drammen.
"Of course it is cool. It is a dream to play in the Premier League one day. Now there was no agreement, so I just try to focus on where I am now."
According to TV2, Burnley had three bids for Arnstad turned down by the Belgian side during the latest transfer window, as Vincent Kompany looked to bolster his engine-room options.
The report adds that the Clarets' final offer was worth in the region of £4.5m, however Anderlecht's stance did not change throughout the negotiations - they did not wish to see the youngster depart.
Arnstad signed a new contract at Lotto Park in the summer of 2022, with the player now tied down to the Pro League outfil until 2025, meaning the Purple and White are in a decent position to drive a hard bargain in future transfer talks regarding the 20-year-old.
With the arrivals of experienced midfielders such as Mats Rits and former Borussia Dortmund man Thomas Delaney, the Norwegian youth international has received limited game time at the beginning of the 2023-24 term.
Featuring in just two of Anderlecht's opening six Pro League contests, Arnstad has started just the single game for Riemer's side so far, playing 68 minutes in the 2-0 defeat at Union SG on July 28.
Kompany's men managed to bring in two central-midfield options in August, with Sander Berge and Han-Noah Massengo joining from Sheffield United and Championship side Bristol City respectively.
The Clarets are still looking for their first points of the top-flight campaign and have conceded a division-high tally of 11 goals in their opening three matches - a trip to the City Ground to face Nottingham Forest is next up on September 18.