The former Liverpool striker turns 33 in January and his Nou Camp deal runs out in 2021.
Rashford was linked with a move to Barcelona before signing a four-year contract, worth up to £300,000-a-week, with Manchester United in July.
Meanwhile, former United and England defender Rio Ferdinand has urged Kane to leave Tottenham in order to win trophies.
Suarez has won four LaLiga titles and a Champions League trophy since swapping Liverpool for Barcelona in 2014.
Spanish outlet Mundo Deportivo claim Barcelona are also keeping tabs on Paris Saint-Germain and France sensation Kylian Mbappe and Inter Milan forward Lautaro Martinez.
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The report adds Barcelona chiefs are ‘more convinced’ by Rashford than Liverpool’s Roberto Firmino, despite his dazzling form for the Reds.
Firmino has netted 19 goals and provided 12 assists since the start of last season.
The Brazilian has helped Liverpool maintain a 100 per cent start to the Premier League this season and an eight-point gap over second-placed Manchester City.