According to Ian Murtagh of the Daily Star, both London clubs admire the centre-back, but Magpies boss Rafael Benitez is desperate to keep him at St James' Park.
It's not hard to see why the 24-year-old has caught the eye of the Premier League giants.
Benitez made Lascelles Newcastle captain at just 22, which speaks volumes about his role as a leader in the dressing room and on the pitch. He remains the youngest skipper in the Premier League.
The Englishman's passion and drive made a refreshing change after several disappointing years on Tyneside, and the Toon Army quickly took him to heart, displaying a flag with his face on earlier in the season, per the club's official Twitter account:
His performances have also earned praise from Newcastle legend Alan Shearer:
Lascelles is a commanding presence at the back for the Magpies, as evidenced by their defensive records with and without him.
With him in the side, Newcastle have conceded 22 goals in as many games this season; in the five matches in which he did not feature, they shipped 14.
He's also a goal threat from corners, having netted three this year—earning the Magpies seven crucial points in the process—and eight overall in 92 outings for the club.
Paul Taylor of the Nottingham Post is a keen admirer of the former Nottingham Forest defender:
His presence could benefit either Arsenal or Chelsea—the former in particular given their defensive struggles and need for strong leadership. And while he's far from guaranteed a place in Gareth Southgate's England squad for this summer's FIFA World Cup, he is perhaps deserving of a call-up given the performances he has put in.
Lascelles signed a new deal in October to keep him in the north-east until 2023, and he's Newcastle's most important player, so he would not come cheap.
Much could depend on the Toon's Premier League survival bid, though, as well as the desire of those above Benitez at the club to bring in a hefty transfer fee for one of their top assets.