The school, based in the former Holmlea Primary building in Cathcart, would have separate entrances and facilities for boys and girls if a fundraising appeal is successful.
The school would offer a range of secular subjects, as well as Arabic and Islamic sciences.
The bid is being led by parents and professionals concerned about the environment in Scottish schools.
Glasgow Community Education Association (GCEA), which is behind the project, claims attending mainstream schools is resulting in "unsocial behaviour" among Muslim youngsters and that the lack of Islamic schools is forcing Muslim families out of Glasgow.
On a website set up to raise money for the project, the association says: "There is a huge demand in the community for a high-standard local facility providing good secular education together with moral guidance in order to produce well-balanced upright individuals."
Some previous attempts to establish Muslim schools in Scotland have been unsuccessful. Both the Iqra Academy in Glasgow and the Imam Muhammad Zakariya School for girls in Dundee closed after receiving negative inspection reports.
The Iqra Academy, which shut in 2003, was criticised by inspectors for giving pupils no opportunity to mix with the local community and for its treatment of girls at the school.
However, more recently the Qalam Academy has been set up in Glasgow, an independent Islamic Educational Institute providing primary education.
Hanzala Malik, Glasgow MSP, disputed the need for a Muslim school.
He said he was a firm believer in public education and urged those behind the school to redirect their energy into improving mainstream provision.
He added: "When children leave school they are all going to have to live in the same world."
Holmlea Primary in Cathcart was declared surplus by Glasgow City Council in 2006 and put up for sale. Six bids were made for it last year, with the GCEA putting in the highest offer.
It describes itself as an organisation formed by "local professionals and parents" to establish a private educational facility for Muslim children. The GCEA envisages it as catering for pre-school to secondary pupils.
The purchase of the school will cost £225,000, exclusive of VAT, and further funds will be required for refurbishment. The GCEA is currently trying to raise the money it needs online.