The TV presenter was ITV's frontman during the early summer football tournament in Ukraine and Poland, with Tate said to have flown out to meet him during the event.
The pair were recently reported to have been discreetly dating for most of the year.
Tate allegedly flew out to spent time with Chiles during the tournament, including a trip to see Auschwitz, reports The Sun.
"Adrian was at the Euros for ITV and she was there just to see him," a source said. "They were very touchy-feely. That's when I realised they were an item."
A friend of Chiles said that the broadcaster is a history buff, explaining: "He is very interested in history, and there are fewer more powerful historical sights than Auschwitz."
Photographs emerged yesterday showing Tate and Chiles on holiday together in Santa Monica, California.
The Office actress Tate, who has a 9-year-old daughter from a previous relationship, is said to have had a relationship with Take That singer Jason Orange in late 2011, before breaking up in February.