Norway midfielder Odegaard was first linked with the Gunners earlier this week as he handed in a transfer request at Real Madrid, and it seems a possible deal has progressed quickly.
According to El Confidencial, the 22-year-old is heading to north London as 'Arsenal will be the Norwegian's destiny'.
Odegaard spent last season on loan at Real Sociedad, scoring seven goals and adding six assists in a fine campaign.
However, he returned to Los Blancos last summer but has failed to establish himself in Zinedine Zidane's team.
He is said to have preferred a return to La Real this month, but the report claims Arsenal have pushed harder for the signing and have as a result won the transfer race.
Additional reports in Spain, from Sport, say Odegaard is only 'one step away' from signing for the Gunners, with the loan deal 'very close' to completion.
talkSPORT's transfer insider Alex Crook joined White and Jordan on Thursday and provided some exclusive info on Arsenal's interest in Odegaard.
"There is genuine interest from Arsenal in Martin Odegaard," Crook told talkSPORT.
"He was a bit of a wonderkid when he joined Real Madrid, he was just 16, but it hasn't quite worked out for him.
"Arsenal would like to bring him in on a loan deal for the rest of the season.
"I wonder if the presence of Dani Ceballos, someone Odegaard would have rubbed shoulders with at the Bernabeu, may help Arsenal get this one over the line.
"There is genuine competition from Real Socieded in Spain."
Martin Odegaard will sign for Arsenal on loan, reports claim.