According to Sky Sports' Keith Downie, the bid was worth £20m, rising to £25m with bonuses and add-ons. However, Newcastle have rejected the bid as 'out of hand', insisting that their captain and star defender is not for sale.
The 24-year-old centre back made 33 appearances in the Premier League for Newcastle last season, chipping in with three goals as Newcastle finished tenth - just ten points outside of the Europa League qualification places.
West Ham now appear to be in the market for Lascelles, having already bolstered their defensive ranks with the £20m signing of 21-year-old centre back Issa Diop from French outfit Toulouse.
Manuel Pellegrini is working hard to build his squad, and Lascelles has built a reputation as one of the best English centre halves in the Premier League largely on the back of his stellar campaign last season, and was widely tipped for a call-up to the England squad for this summer's World Cup in Russia.
Speaking as speculation built around Lascelles' future back in April, manager Rafael Benitez was full of praise for Lascelles and insisted that his captain is 'happy' at the club.
"He's happy, training well, he's improving, he's focused," Benitez said.
"You can't stop speculation about players, but I was talking with him for a while last week and the focus is just trying to improve his game.
"I think at the moment he knows he has to continue progressing and learning, but he is quite happy here."
Newcastle have signed two players so far this summer, with loanee goalkeeper Martin Dubravka being the first to join the club on a permanent deal. The club have also signed midfielder Ki sung-yueng on a free transfer from Swansea.