Both clubs have been linked with a move for the 22-year-old this summer, who is unlikely to see much first-team action ahead of the likes of Toni Kroos and Luka Modric in La Liga.
He is highly rated at the Bernabeu though, with former manager Carlo Ancelotti having used him in parts last season, and Armstrong says they are unlikely to sell, with a loan deal to a Premier League club the more likely option.
“It's claimed Manchester United and Arsenal are in a race to sign Real Madrid midfielder Lucas Silva,” he wrote on Kicca.
“I can't imagine Real selling him as he is very much seen as a player for the future. He may well join a Premier League club but only on a loan deal to give him some much-needed game time and match experience.”
Arsenal or Tottenham could be interested
With Armstrong suggesting a loan move for game time and experience is more likely, several Premier League clubs may be interested.
Arsenal would likely be keen on a deal as it would give them suitable competition in their defensive midfield area, with Francis Coquelin in need of solid back up in the side.
Rivals Tottenham Hotspur could also be interested in a deal. It has often been suggested that they need a defensive midfielder in their side, and Silva would be a good option for them in a loan deal.
Either way, it seems Silva may be plying his trade in the Premier League next season.