Chris Pratt during an interview at the Fellow Bar in Los Angeles. Picture: AP

Chris Pratt and Katherine Schwarzenegger are in a "serious relationship", but are still taking things day by day.

The 39-year-old actor has been romancing Katherine, the daughter of Hollywood icon Arnold Schwarzenegger, since June, and sources have said that although the romance is heating up and becoming increasingly more serious, Chris isn't likely to be getting down on one knee any time soon.

An insider told 'Entertainment Tonight': "[They are in a] serious relationship and everything is going well. However, Chris just finalised his divorce from Anna [Faris] a couple months ago, and he's happy with just dating Katherine for now. 

"The couple has a lot of fun together and is fully involved in one another's life. [They are] in a committed relationship. There just isn't a rush to take it to the next step right now. Chris is enjoying the dating process."

The 'Guardians of the Galaxy' star - who has a six-year-old son Jack with Anna - is even said to have won over Katherine's father Arnold, after the three met up with Arnold's girlfriend Heather for brunch over the weekend.

One source said: "Chris and Katherine met up with Arnold and his girlfriend Heather for Sunday breakfast at the FIG restaurant. Arnold seemed happy that Chris and Katherine joined him."

Whilst another insider added: "A lot of laughter and storytelling. Arnold seemed very pleased to spend time with Chris. It was obvious that they got along and enjoyed getting to know each other. They had eggs, breakfast meats and fruit. Chris settled the check."

Meanwhile, Chris and Katherine are said to be "inseparable".

An insider shared: "The moment Chris and Katherine started dating, everything fell into place and they seemed inseparable. They just love each other's company and have so much in common. They spend a great deal of time together. Chris and Katherine have the full support of her family and Maria [Shriver] feels they are a match made in heaven."