First Minister Mark Drakeford has said the Welsh Government is prepared to increase fines for those breaking coronavirus lockdown laws.
It comes as police officers across the country issued fines throughout the weekend to those travelling within Wales and from England.
Speaking at the Welsh Government's daily coronavirus press conference, Mr Drakeford said "If the evidence demonstrates that there is a problem that can be solved and should be solved by raising the levels of fines in Wales, then of course we are prepared to discuss that with our Chief Constables."
In Pembrokeshire, Dyfed-Powys traffic officers said they had made four arrests, submitted 12 traffic offence reports and that 26 people had been reported for non-essential travel.
In Powys, ITV Wales journalist Hannah Thomas spent time with officers who stopped a family of four from Birmingham visiting Pen y Fan and a driver from Devon travelling to buy dog food.
The First Minister said these examples were "unacceptable" in the same way it is "unacceptable for people who live in Wales to travel distances to other parts of Wales."
Fines in Wales currently begin at £60, and rise to £120 if people have already received a fixed penalty notice.
However, fines in England now start at £100 and can reach up to £3,200 for repeat offenders.
In Wales, lockdown regulations state that people must exercise locally.
The rules in England however have been relaxed to allow people to drive to other destinations to exercise and meet one person outside of their household.