The Belgian international is a late target for the English Premier League side.
Nottingham Forest had previously tabled two offers for the midfielder, amounting to £14 million and £17.3 million, which were also rebuffed by the club.
Celtic are said to have made the decision to decline the £20million bid, which could have risen to £25million with add-ons, despite the chance to double their money on a player signed two and a half years ago from Augsburg.
The Parkhead side are reluctant to sell in the current window.
The Scottish club are on the verge of completing deals for loan striker Junior Aduma from Austria, and Norwegian winger Joel Mvuka.
The sale of Engels with no replacement lined up would leave manager Martin O’Neill short of midfield options heading into a tense title run-in.
The English window closes at 7pm, with the Scottish deadline four hours later.