I can't answer this claim directly, but it is fairly certain that the Mongols sent scouts to check the defenses of at least the Holy Roman Empire and Italy just after the Mongol invasions of Hungary and Poland in 1241-42, to plan for the next planned invasion that never occured. This wasn't under Ghengis Khan though, but under his son Ogodei. That the Mongol leaders Subotai and Batu Khan should send scouts further to France and to England is not implausible, but I'm not sure its confirmed.

The Mongol (actually a Nestorian Christian Uighur) ambassador Rabban Bar Sauma who visited Europe in the 1280's met with king Edward of England in Gascony in southern France, which was an English possession at the time, in 1287, which was a favorable meeting, but I think that is the full known extent of the Mongol-English connections.

In general, countries fearing the Mongol invasions in that era were prone to paranoia and seeing Mongol spies everywhere, regardless of how likely they were to actually exist. As an example, during the Mongol invasions of Japan, people were executed all over Japan, including one in Sendai to the far north, due to highly unlikely suspicions of them being Mongol spies. The Mongols had plans for invading Europe a few times, but these never got beyond Eastern Europe, and it is unlikely they would have had spies in places that weren't their next immediate targets, such as Germany and Italy.

This is just speculation, but I have reason to believe it to be the truth - I'd consider Mongol spies in England specifically to be unlikely.

Do you remember where you read that?