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Nucleophilic substitution through an elimination-addition mechanism (involving benzyne formation) occurs with a strong base. $\ce{NH2-}$ can show this reaction. Can Grignard reagents $\ce{RMgX}$, which form $\ce{R-}$ - an even stronger base - show this reaction?

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    $\begingroup$ Griniard reagent doesn't form R- and usually acts as nucleophile not base $\endgroup$ – Mithoron Jun 30 '15 at 12:32
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    $\begingroup$ Actually a Grignard reagent can act as a (protic) base. Expose it to water and it's bye-bye Grignard reagent as the "R" part gets converted to a hydrocarbon. Chemists handling Grignard reagents must exclude water and anything else with which the reagent can react as a base. $\endgroup$ – Oscar Lanzi Mar 2 '16 at 2:23
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NAS reactions normally require strong electron withdrawing groups to facilitate the addition. Normally these groups, nitro, carbonyl, cyanide, are preferentially good electrophiles for Grignard reagents (nucleophiles).

So no, organomagnesium reagents will not react via NAS.

You also mention the Benzyne mechanism (which is somewhat different). In this case, where nucleophilic attack on other groups is not possible, I expect metal-halogen exchange due to aryl anions being slightly weaker bases than alkyl anions. After said exchange benzyne formation is possible.

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