Gunners midfielder Ramsey had his leg shattered in six places in 2010 after a tackle by Shawcross and refused to accept the centre-half's attempts to apologise.
Arsenal manager Arsene Wenger described the Shawcross challenge as "horrendous and unacceptable" and there has been significant tension between the clubs since then.
But Coleman is keen to bring Shawcross into his squad as the 24-year-old English-born player qualifies for Wales because he spent five years of compulsory education in the principality.
Shawcross, who began his career in Manchester United's youth academy, rejected a previous approach to play for Wales.
But Coleman is convinced he can change his mind and he will open talks once he hears that Ramsey is willing to make peace with his rival.
"Ryan is a good player who could strengthen us," Coleman told The Sun.
"I will have a conversation with Aaron because he's somebody I respect a great deal, he's our captain and an important player.
"I won't shy away from it because there are issues."
Coleman also wants to recruit Swansea's Spanish-born defender Angel Rangel and Cardiff defender Ben Turner, who qualifies through Welsh grandparents, as he prepares for the start of the 2014 World Cup qualifiers in September.
Chris Coleman will consult Aaron Ramsey over Ryan Shawcross talks
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