According to Iraqi police and hospital officials, the blasts occurred on Sunday near the provincial council building in Ramadi, Anbar's capital, 70 miles (115 kilometers) west of Baghdad.
The first explosion was heard at 10 a.m. (0700 GMT), killing at least 17 people, among them six members of the police force.
At least 23 people have also been left wounded in the Sunday attack.
Only minutes later, a second bomb was found in a nearby parking lot. Police say they have detonated it in a safe area.
No group has yet claimed responsibility for the attack.
Government officials in Anbar, a former al-Qaeda stronghold, have frequently been targeted by militants.
The violence comes days before the climax of the Shia commemoration ceremony of Ashura on Thursday.
The tenth day of Muharram, which is known as Ashura, marks the anniversary of the martyrdom of Prophet Mohammad's grandson, Imam Hussein (PBUH).
Every year on Ashura thousands of Muslims from all over Iraq and neighboring countries hold mourning ceremonies and many head for the holy city of Karbala, where the shrine of Imam Hussein (PBUH) is located.
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