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offered to do和offered to doing都可以。Offered to do主动提出做某事；Offered doing 提议做某事。是动词的两个不同的发展阶段。凡是需要表示还未发生的那么就接动词 to do， 凡是已经发生，或者正在发生的，那么就用动词doing。
offer的用法
1)offer作动词(verb)
(1)offer sth (to sb) (for sth) | offer sb sth 主动提出;自愿给予
【例句】
Josie had offered her services as a guide.
乔西曾表示愿意当向导。
He offered some useful advice.
他提出了一些有益的建议。
(2)offer sth/sb (up) (to sb) 奉献，祭献(给上帝)
【例句】We offered up our prayers for the men's safe return. 我们祈求上苍保佑他们平安归来。
2)offer作名词(noon)
(1)offer (of sth/to do sth) 主动提议;建议
【例句】I took him up on his offer of a loan. 他主动借钱给我，我接受了。
(2)offer (for sth)
【例句】They've decided to accept our original offer. 他们已决定接受我们最初的报价。