A man reportedly chased two women near the Tesco Extra supermarket in Yate.
An Avon and Somerset police spokesman says on both occasions the man was spotted hiding in bushes on Kennedy Way.
He said: "A woman reported a man who had been hiding in bushes chased after her at about 9.10am on Friday, January 15.
"She ran into the supermarket car park to get away, later reporting it to police through the website and posting about it on Facebook.
"A second woman saw the Facebook post and called police the following day to report a similar incident which happened to her at about 6.50pm on Tuesday, December 1.
"Officers urge anyone who is fearful for their personal safety due to the behaviour of a stranger to head for a busy public area, such as a shop, and call police straight away."
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Both women described the "balding" man as being in his 40s or 50s.
One said he was wearing black trousers and a blue coat. The other described him as about 6ft tall and slim.
The force spokesman added: "Were you travelling along Kennedy Way at those times? Do you have any dashcam footage which could help?
"If you have any information call 101 quoting reference 5221011297.
"Alternatively ring the independent charity Crimestoppers on 0800 555 111. They never tell the police who you are, just what you know."
