Many simply stay put or move to another country within the bloc, she said.
Member states should all recognise the decisions taken by other EU countries to ensure that “migrants who have a return decision against them in one country cannot exploit cracks in the system to avoid return elsewhere”, Von der Leyen wrote.
Her comments come as Italy kicks off its long-awaited scheme, under which some of the migrants rescued in the Mediterranean will be sent to Albania for processing.
Earlier this week, 16 men of Bangladeshi and Egyptian origin were moved from the migrant hotspot of Lampedusa, off the coast of Sicily, to one of two purpose-built centres on the Albanian coast where their asylum claims will be examined.
The centres, which cost about €650m (£547m), were due to open last spring but were plagued by long delays, have been paid for by the Italian government and will be operated under Italian law.
They will house migrants while Italy examines their asylum requests. Pregnant women, children and vulnerable people will be excluded from the plan.
Political opponents of right-wing Italian Prime Minister Giorgia Meloni as well as several NGOs have criticised Italy’s deal with Albania.
Riccardo Magi, an MP with the left-wing +Europa party, said the Albania scheme was “cruel, useless and expensive”, while NGO Doctors Without Borders said it was “likely to result in further harm and violation of human rights”.
However, addressing MPs on Tuesday, Meloni argued that the plan was “a new, courageous, unprecedented path” which “perfectly reflected the European spirit”.
The implementation and the results of the Albania agreement will be watched closely by many EU member states, several of whom have attempted to respond to a surge in support for far-right parties by hardening their rhetoric and their approach to migration.
In the last few weeks alone, Germany reintroduced land border checks, the French government said it would look into tightening immigration legislation and Poland announced a plan to temporarily suspend the right to asylum for people crossing the border.
Polish PM Donald Tusk said the controversial move was meant to stop Belarus from “destabilising” Poland by allowing large numbers of migrants into the country.
In France and Germany, it was grisly murders which prompted calls for tougher action on immigration. A Syrian failed asylum seeker stabbed three people to death in Solingen, while a young student was murdered by a Moroccan national near Paris. In both cases, the killings were carried out by men who had been given expulsion orders that had not been enforced.
Last month, 15 member states signed a proposal by Austria and the Netherlands to improve the “efficiency” of the deportations system.