A media spokesperson for the Algerian FA declined to confirm the report.
"We are dealing confidentially with the issue and will not talk about it. We will leave coach Djamel Belmadi with the task of revealing the new team," Saleh Bey Abboud told reporters.
Weiser is eligible to represent Algeria because of his mother's origin, Kicker said.
Weiser participated in 40 matches with German youth teams, and won the European Championship title with the under-21 team in 2017, but did not get a call-up for the first team.
The Algerian Federation has already contacted the 28-year old right-back, but joining the team depends on the plans of coach Djamel Belmadi and how fast the formal procedures for obtaining citizenship can be formalised.
Weiser, who is the son of former player Patrick Weiser, made an impressive start to the season, assisting in seven goals in his first 14 games before the World Cup in Qatar.
But that wasn't enough for coach Hansi Flick to include the former Bayern Munich player in the German national team.
Belmadi led Algeria to their second Africa cup of Nations title in 2019, but the team were eliminated in the first round in Cameroon early last year.
Algeria also failed to reach the World Cup finals in Qatar as they lost 2-1 in extra-time to Cameroon on home soil after winning 1-0 in Douala.
Coach Belmadi promised to rebuild the team to compete for the Nations Cup title in Côte d'Ivoire in 2024 and qualify for the 2026 World Cup.
Algeria tops Group F in the Nations cup qualifiers with six points from two matches, and will face Niger on March 23rd and 27th.
