EIB president Werner Hoyer told a press conference that the situation will be in "limbo" for two years until Brexit talks are complete, but that the EIB is keen to continue to work with the UK.
The Brexit vote was a "terrible shock for us, we regret it very much", Hoyer said. The UK is a major shareholder in the bank, with 16.1% of shares in British hands.
The UK, in return, relies heavily on the EIB, and "we will be missed if we have to reduce our business there, or disappear", he said.
"The EIB invested over €7 billion in the UK last year. That is enormous," Hoyer said.
While the statutes of the bank make it clear that the shareholders of the EIB are member states of the European Union, whether "there is an option to change that and allow a former member state to remain a member … is completely open to speculation. So I think we should not exclude any possibility", he said.
In the meantime, Hoyer said, "we will continue to do business with the UK. The UK will remain a shareholder for at least two more years and we have good projects and very solid relationship."
Banking expert Tony Anderson of Pinsent Masons, the law firm behind Out-Law.com said: "This is another example of the unconsidered and far reaching ramifications of the referendum result. It would be hoped that the rules could be altered to accommodate the UK as a continuing shareholder. A significant drop in investment into the UK from the EIB would be keenly felt."
In October, Hoyer warned that recent levels of lending to UK infrastructure projects by the European Investment Bank (EIB) cannot be maintained as the country moves towards leaving the EU. The bank does lend to non-EU countries but at lower levels, and lending will have to be reduced even before the UK actually leaves the EU, he said at the time.
The EIB was set up in 1958 under the Treaty of Rome. It describes itself as "the European Union's bank", and "the only bank owned by and representing the interests of the European Union member states. We work closely with other EU institutions to implement EU policy."
The EIB approved €3 billion in new financing for 17 projects across Europe last week, including clean transport and energy initiatives.
The projects are backed by the EU budget guarantee under the European Fund for Strategic Investments (EFSI).
The EFSI was established last year by the EIB as a joint initiative of the European Commission and the EIB. It will manage a guarantee from the EU budget of €16bn and an EIB contribution of €5bn to trigger private and public investment of €315bn.