According to reports in English media, Rodriguez, who's fallen out of favour at the Bundesliga giants and Los Blancos has £3m left on his overall £11.5m loan payment, which Unai Emery's side could stump up to lure him to The Emirates for the final six months of the 2018/19 season.
And the boss may have received a huge boost as The Mail claim he is available at a knockdown price.
Rodriguez signed a two-year loan deal at Bayern from Real Madrid which expires at the end of the season.
He was signed by former manager Carlo Ancelotti but has fallen out of favour since the Italian was sacked and replaced by Niko Kovac.
The Gunners are struggling to raise funds for permanent transfers in the winter window so it could represent an ideal solution for both clubs.
Fiscal de Perú acusará a Humala y Heredia por lavado activos
En enero de 2015, se inició la investigación a Nadine Heredia Alarcón luego que la Unidad de Inteligencia financiera detectara transferencias por 216 mil dólares a cuentas bancarias de la entonces primera dama.
If Arsenal can sell him off in time then they would finally get the money to bring in the much-needed signings.
The situation has alerted Arsenal with the Daily Mail claiming that we've identified Rodriguez as a potential addition this month as Unai Emery looks to strengthen his squad on a shoe-string budget.
The 27-year-old has made just eight league appearances for Bayern Munich this season, but has totalled under 500 minutes of on-pitch action, contributing three goals and a single assist.
He won the Golden Boot at the 2014 World Cup where he inspired Colombia to the quarter-finals and was named in the team of the tournament.