"If we don't see anything happen from the Turkish side in the next few weeks and the number of refugees does not reduce significantly, Europe must put the agreement to the test," Weber told the Suddeutsche Zeitung.
The politician is the leader of the largest political group in the European Parliament, the European People's Party (EPP), a coalition of center-right political parties. Weber himself is a member of Angela Merkel's Christian Democrats.
Though in recent days the number of refugees arriving in Greece each day has fallen to around 1,500, Weber said that EU members have attributed the reduction to bad weather rather than the efforts of Turkish authorities to stem the influx of refugees and migrants into Europe.
Flüchtlinge — Enttäuschung über die Türkei https://t.co/f7CCAK4oTC— petra (@kuppp005) 8 января 2016
'Disappointment about Turkey,' reported the Suddeutsche Zeitung on Friday.
"The results are disappointing," he said, but remarked that criticism leveled at Turkey should be accompanied by corresponding criticism for EU member states who are also failing to control their borders.
"We have to set an example ourselves," said Weber.
In addition, Turkey received political concessions including the promise of visa-free travel for Turkish citizens in Europe and the renewing of talks on its accession to the EU.
However, EU representatives are so far disappointed with the outcome of the agreement. SZ reported Vice-President of the EU Commission Frans Timmermans as saying that "we are far from being satisfied" about the steps taken by Turkey to stop illegal migration.
On Thursday Greek Migration Minister Yannis Mouzalas complained that between 3,000 and 4,000 migrants and refugees are still reaching Greece every day.
"It (Turkey) has not done anything to stop human trafficking, as is evident from the migratory flows," Mouzalas told Greek television.