It’s understood Accies value the 21-year-old at £1million plus add-ons.
Cardiff City are also keen on Docherty, who has come through the youth ranks at New Douglas Park.
Rangers boss Graeme Murty has already brought in midfielder Sean Goss on loan from QPR and Jamie Murphy in a similar deal from Brighton.
Meanwhile, Murty believes the number of top players keen to sign for Rangers proves that the club remains a major draw in European football.
Scotland striker Stevie Naismith is desperate to return to Ibrox, but Murty admitted he has several targets.
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He said: “We are working hard behind the scenes on a number of things.
“You forget the regard the club is held in around England and Europe, because some of the people who are phoning up are high level.
“It has been great for me to have a sounding board in Mark Allen [director of football] to guide me in some regards and get me prepped for a busy period.
“The idea is to strengthen the group and take us to a higher level than we are currently operating on.”
There could be players leaving the club to make way for more new arrivals at Ibrox this month.