Johnson heads to Brussels on Wednesday for last ditch trade talks
The pair are under pressure to get an agreement before the UK stops following EU trading rules on 31 December.
The dinner meeting comes after intensive talks between EU and UK officials ended in deadlock.
Major disagreements remain on fishing rights, business competition rules and how a deal will be policed.
Mr Johnson will work through a list of the major sticking points with Mrs von der Leyen, who is representing the leaders of the 27 EU nations.
He will take part in Prime Minister's Questions before travelling to the Belgian capital. EU leaders are due to meet for a summit of their own on Thursday.
Chief UK negotiator Lord Frost said Wednesday evening's meeting would allow the two sides to "continue our discussions" over a future trade deal.
EU sources told the BBC his EU counterpart Michel Barnier briefed the bloc's Europe ministers that talks were tilting towards no deal being reached before the deadline.
Earlier, Mr Johnson said he hoped the "power of sweet reason" will allow both sides to clinch an agreement before the end of the year.
If an agreement is not reached and ratified by this date, the UK and EU could introduce import charges on each other's goods.