The newspaper also reports that Liverpool are interested in signing the Norwegian, but are unwilling to match his £80,000 per week salary.
The 17-year-old moved to the Spanish capital in January 2015 from boyhood club Stromsgodset for a fee believed to be in the region of £3 million.
The Sun claim that East London is the most likely destination for the Madrid teenager, with the Hammers' need to bolster there squad becoming more urgent after midfield maestro Manuel Lanzini was ruled out for five weeks with a knee injury (The Standard).
West Ham could certainly use more strength in depth in the attacking midfield area, and seem to lack any kind of creativity when Lanzini and Dimitri Payet are absent from the starting XI.
A season in the Premier League would be extremely useful for Odegaard and would go a long way to assist in the development of his game - from a physical aspect at least.
With just over three weeks remaining before the transfer window slams shut, West Ham are expected to be one of the more actively involved clubs, and if a move for Odegaard does come to fruition, albeit on a temporary basis, it will certainly be one to get the Hammers fans excited.
Will West Ham manager Slaven Bilic make a move for Odegaard?