The company approached network agencies last week and will hold chemistry meetings with shortlisted agencies later this month.
The pitch is understood to have been called as part of a renewed effort by Avis to get more travellers to use the brand on a regular basis.
Avis is under pressure from rival car rental companies such as Europcar, Sixt and Hertz in a crowded market, and is keen to implement a stronger loyalty programme to help increase customer retention.
Last year, Avis appointed Acknowledgement and Web Liquid to handle a pan-European direct marketing campaign that positioned the brand as a value-for-money, premium rental service.
The two agencies were awarded the work after a pitch against Rapp and McCann Erickson, the previous incumbent on the direct marketing account.
The campaign was rolled out in 13 European markets, including the UK, Germany, Spain and Italy, and featured the brand's famous "We try harder" positioning, which was introduced by DDB in 1962.
Web Liquid and Acknowledgement also handle Avis' affiliate marketing and digital display activity. These relationships are not affected by the pitch process.