Sky Sport reports that representatives of both clubs were in attendance to assess the 20-year-old centre-back, who has rapidly developed into an important player for Stuttgart.
Jeltsch has attracted wider attention following his rise from 1. FC Nürnberg to the Bundesliga, with his performances also earning him his first call-up to the Germany senior squad.
Barcelona's interest comes as the Catalan club continues to examine young options for its centre-back positions. The latest report indicates that Jeltsch is being monitored rather than suggesting that negotiations have begun.
Inter Milan are also following the defender, while Sky Sport says several Premier League clubs have been keeping track of his development.
VfB Stuttgart sporting director Fabian Wohlgemuth declined to discuss the specific scouting activity when questioned by Sky Sport, although he highlighted Jeltsch's rapid progress and potential.
The defender has already been given major responsibility in Stuttgart's back line this season. He started the Champions League meeting with Viking, which Stuttgart won 3-1, and has become one of the club's most closely watched young players.
Jeltsch's rise has been particularly rapid since his move from Nürnberg. His performances in Germany's top flight have transformed him from a promising young defender into a player attracting attention from clubs across Europe.
Real Madrid have also previously been linked with the German, adding another major club to the growing list of teams monitoring him.
For Barcelona, the appeal is clear. Jeltsch is young, already playing regularly at a high level and now beginning to gain international recognition with Germany.
No transfer negotiations have been confirmed, with the current interest focused on scouting and assessment. The Champions League appearance nevertheless provided Barcelona and Inter with an opportunity to watch the defender directly.