Ramsey is currently in the final year of his contract at the Emirates Stadium, and the agency representing him recently appeared to confirm that he would be leaving the club either in January or when his deal expires next summer.
Arsenal had been keen to tie the Wales international down to a new deal, but the two parties are understood to have been significantly apart in talks over wages and the negotiations have now broken down.
The Gunners are already considering replacements for Ramsey, and the Evening Standard reports that they have identified Herrera as a possible target in the January transfer window.
Herrera has spent the past five years with Porto and also represented Mexico at this summer's World Cup in Russia.
Ramsey, meanwhile, has been an Arsenal player since 2008 and has attracted interest from the likes of Liverpool, Manchester United and Juventus amid his ongoing contract standoff.