Officers have been searching for the 36-year-old, from Petworth in West Sussex, since 22:00 BST on Thursday.
Ms Weeks, who is known for her role in the ITV drama, was last seen an hour earlier 14 miles away in Chichester.
Sussex Police said they were concerned for her welfare and it was unlike her not to get in touch. She had recently told family and friends she was feeling anxious.
Det Kate Wilt said: "If you read this Honeysuckle, please get in touch to let us know you are ok.
"Although she travels around a lot and has links in London and has family in Wiltshire, it is unlike her not to be in touch with family."
Earlier this year the actress was reportedly ordered to wear an electronic tag after she was caught speeding on the A3 in south-west London in August 2015.
A child in the back seat was not wearing a proper seat belt and it emerged she was already banned from driving.
Her agency The Artists Partnership said it was currently "unable to get hold of Honeysuckle".
She recently finished work on shooting Lewis for ITV and a Sky mystery The Five.
As well as starring as Samantha Stewart in drama Foyle's War, she has also appeared in The Bill, The Inspector Lynley Mysteries and Death In Paradise.
The Cardiff-born actress is described as around 5ft 4in (1.62m) with cropped gingery blonde hair.
She was last seen wearing a blue anorak and faded blue jeans.
Foyle's War actress Honeysuckle Weeks has gone missing, police have said.